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c4    (^    2f^A,*'Hvnvv^ 

EM 


JS  Y  S  T  E  M 


OF 


'"  FOR 

THE   USE    OF    SCHOOLS, 

BY  THE  REV,  J.  JOYCE. 


ASAPTU)   TO  THE 


COMMERCE  OF  THE  UNITED  STATES, 
BY  J.  WALKER. 

BALTIMORE : 

PUBLISHED  BY  N.  G.  MAXWELL,  140,  M ARKET^TREE  1 

J.  Robinson,  Printer. 

1819. 


DtSTRlCT  OP  MAnTLAITD,  SS. 

BE  IT  liEMKMBERED,  That  on  the  Twenty-First  day  of  June 
,*:Mai**»   »"  the  Forty-third  year  of  the  Independence  of  the  United 
IskalI  ?t^tes  ot  America,  X  (i.  Maxwell  ofthesaid  District  hath 
%„^^^  deposited  m  thisofllce  the  title  of  a  Book,  the  right  where- 
«  A  c    *   '^^  he  Claims  as  Proprietor  in  the  words  followinP  to  wit: 
«  l^et    7  i^""  ""^  iVactical  Arithrnetick.  for  the  Use  of  Schools,  by  the 
"  J    \Vaike?''^    Adapted  to  the  Commerce  of  the  United  Stites,  by 
In  conformity  to  an  act  of  the  Congress  of  the  United  States,  enU- 
tied.      An   act  for  the  encouragement  of  learning,  by  securing  the 
cop,esof  nxaps,  charts,  and  books,  to  the  authors  and '  proprSrs  of 
such  cop.es  during  the  times  therein  mentioned."    And  ilso  to  the  act 
em.tled,  "An  act  supplementary  to  an  act,  entitled,  «Anacrfor?he 
encouragementof  learning,  by  securing  the  copies  of  maps,  ch^t  and 
books,  to  the  authors  and  proprietors  of  such  copies  during  the  times 
therein  mentioned,"  and  extending  the  benefits 'thereof  t? the  arts  of 
designing,  engraying,  and  etching  historical  ami  other  printe  " 

PHILIP  MOURE, 
•Clerk  of  the  District  of  Maiyljind. 


PREFACE  TO    THE  ENGLISH    EDITION. 


IN"  presenting  a  new  System  of  Arithmetick  to  the 
publick,  some  account  of  its  plan  and  execution  will  be 
expected.  It  is  hoped,  that  the  title  of  the  present  Work 
will  briefly  explain  the  views  of  the  Author,  who,  from 
his  own  experience,  in  the  business  of  education,  has 
long  since  been  convinced,  that,  among  the  excellent 
introductory  books  to  this  useful  science,  no  one,  that  he 
has  met  witii,  is  sufficiently  adapted  to  the  occasions  of 
Gominon  life  :  some  are  too  abstruse  for  novices,  while 
others  are  defective  in  such  examples,  as  point  out  the 
application  of  the  several  rules  to  transactions  of  real 
business. 

If  the  Author  of  this  System  of  Arithmetick  has  not 
deceived  himself,  he  has  completely  supplied  these  de- 
ficiencies, and  he  appeals  without  apprehension  to  that 
publick,  whose  candour  and  liberality  he  lias  already  and 
often  experienced,  to  decide  upon  this  attempt  to  rendei 
the  elementary  rules  of  arithmetick  practical  and  po- 
pular. 

There  are  few  children  who  do  not  experience  some 
disgust  in  passing  through  the  first  four  rules  ;  occasion- 
ed, without  doubt,  by  the  paucity  of  examples,  and  by 
the  want  of  interest  in  those  that  are  given.  The  Au- 
thor has  therefore  filled  a  large  portion  of  his  work  with 
the  early  rules,  and  has  illustrated  them  by  miscellane- 
ous questions,  in  which  will  be  found  much  useful  infor- 
mation, applicable  in  the  advancing  stages  of  life. 

The  modes  of  treating  the  Rule  of  Three,  of  illustra- 
ting Vulgar  and  Decimal  Fractions,  Practice,  &c.  &c., 
will  best  speal^  for  themselves.  But  a  reason  may  be  de- 
manded for  the  introduction  of  Logarithms,  and  for  the 
particular  method  adopted  in  those  parts  in  whici:  the 
doctrine  of  Annuities,  Reversions,  Leases,  &c.  is  illus- 
trated. 


ARITHMETICK, 


ARITHMETIC K.  is  the  science  which  explains  the  va.- 
riou^  methods  of  computing  by  numbers. 
All  its  operations  are  performed  by  Addition,  Sub- 
traction, Multiplication  and  Division. 


OF  NUMERATION  OR  NOTATION. 

When  two  or  more  figures  are  placed  together,  the 
first  or  right  hand  figure  is  taken  for  its  simple  value  : 
the  second  to  the  left  signifies  so  many  tens  :  the  third  so 
many  hundreds ,-  and  the  fourth  so  many  thousands  / 
and  soon,  according  to  the  following  Tabie  : 

^^      ^  ;§      jg^      H:g      ^      53      ^      t) 

5  431  26978 

Thus  figures,  besides  their  common  value,  have  one 
which  depends  upon  the  place  in  which  they  stand  when 
joined  to  others;  6  and  5  are  read  six  and  five;  but 
if  they  stand  together,  65,  they  are  read  sixty-five. 
The  figure  5  on  the  right-hand  denotes  its  simple  value 
only,  hut  the  6,  from  its  situation,  becomes  ten  times 
greater  than  its  simple  value,  or  sixty,  therefore  the  two 
together  are  called  sixty -five. 


10  NUMERATION, 

If  there  be  three  figures,  as  978,  the  first  figure  to 
the  rijfht-hand  denotes  its  simple  value,  as  eij^ht ; 
the  second  a  value  ten  times  j^reater  than  its  sinple  va- 
lue, as  seventy ;  and  the  third  is  a  hundred  times  5;reater 
than  its  simple  value,  as  nine  hundred  :  the  figures  to- 
gether are  read  nine  hundred  and  seventy-eight. 

In  this  manner,  the  value  of  each  figure  to  the  left  is 
always  ten  times  oreater  than  it  would  be  if  it  stood  in 
the  next  place  on  the  right ;  thus  6666,  the  first  figure  6 
is  simply  six,  the  next  is  sixty,  and  the  third  six  hun- 
dred, and  the  fourth  six  thousand  ;  the  whole  number  is 
read,  Six  thousand  six  hundred  and  sixty -six. 

The  first  six  figures  in  the  table,  are  read,  One 
hundred  twenty-six  thousand,  nine  hundred  and  seven- 
ty-eight. The  whole  period  of  nine  figures  is  thus  read. 
Five  hundred  and  forty-three  millions,  one  hundred  and 
twenty -six  thousand,  nine  hundred  and  seventy -eight. 

The  enumeration  of  figures  may  be  carried  much  fur- 
ther, according  to  the  following  Table  : 


IE  o  5  fi 

t^  .£    .  ^f  .2  «    .  -S 

■^^S-5<«o  -^-^s-o^.S  -acrs-Oci;* 

«     .~4)_n3a.:^        3a)-C3a;i-^  3a).c3a;S 

KHHXHCQ  Er-r-XH^  KHr^EHlD 

123456       4-87951  4627  5  3 

In  large  numbers  it  is  common  to  divide  them  into 
periods  of  six  figures  each,  and  half  periods  of  three  fi- 
gures. The  foregoing  three  periods  are  read — One  hun- 
dred twenty-three  thousand,  four  hundred  and  fifty-six 
billions,  four  hundred  eighty -seven  thousand,  nine  hun- 


NUMERATOIN". 


n 


dred  and  fifty-one  millions,  four  hundred  sixty-two  thou- 
sand, seven  hundred  and  fifty-three.* 

Hence  the  following 

Rule.  To  the  sim/ile  value  of  each  figure,  join  the 
the  name  of  its  place  according  to  the  situation  in  the  se- 
ries, as  hundreds,  thousands,  millions,  billions,  trillions^ 


EXAMPLES  IN  NUMERATION    AND    NOTATION. 


Read,  or  write  down  in  words,  the  value  of  the  fol- 


owing  Numbers 

Sx.  1.  19 

Ex.  11. 

40005 

Ex.  21. 

340 

2.  244 

12. 

324060 

22. 

436i:0l 

3.  3045 

13. 

4(>0569 

23. 

36945 

4.  45060 

U. 

765 

24. 

9874000 

5.  69305 

15. 

564001 

25. 

6.54328 

6.  93614 

16. 

439762 

26. 

4328764 

7.  564875 

17. 

9300044 

27. 

856540 

8.  4500342 

18. 

70000021 

28. 

4376(<O0O 

9.  5687041 

19. 

35000 

29. 

37004 

10.  6843700 

20. 

50000000 

30. 

85L00341 

•  The  names  of  the  higher  periotlt  after  JiillionSf  are  Trillions. 
Quadrillions,  Quintiilio/is,  SextilUons,  SeptiUiunSt  Octillions,  and 
J^'ottiiiions,  each  period  consisting  of  six  places  of  figures.  The 
first  three  of  every  period  are  so  many  Units  oj  it,  and  the  latter 
or  left  hand  part,  so  many  Thousaudi. — The  following  Table  con- 
tains the  whole  series ; 


Nonillions, 

123.456 


Octillions, 
45(.7b9 


liuadrillions,    Trillions, 
674,321         374,532 


TABLE. 

Septillions, 
567,345 

Billions, 
459,876 


Sextillions, 
321 ,234 


Millions, 

53J,761 


Quint  ill  ions 
458,764 

Units. 
459,579 


12  NUMERATION, 

Ex.  31.  55fi074328 
32.  590U007643 
S3.  686  "0749004 

34.  87643078t)45S 

35.  10O0OOO84S218 

36.  3487b5432 18764 

37.  594632171834765 
38    87643ii85ir6487589 

39.  1234o67890012.i9 

40.  987654321123456789 

Write  down  the  figures  answering  to  the  following 
£xan)ples. 

Ex.  1.  Thiriy-nine. 

2.  Four  hundred  and  sixty-nine. 

3.  Two  thousand  and  one. 

4.  Thirty-fivp  thousand  and  twenty-eight. 

5.  Three  hundred  and  seventy-six  thousand. 

6.  One  million  and  fifty-nine. 

7.  Eighty-seven  millions,  five  hundred   and   eighty 

thousand,  one  hundred  and  nine. 
S.  Five  hundred  seventy-six  millions,  three  hundred 
twenty-five  thousand,  three  hundred  and  nine- 
ty-one. 
9.  Eight  hundred  millions  and  ei^thy. 
10.  Three  hundred  and  three-millions  and  thirty-^ 
one. 

MISCELLANEOUS    F.XAMPLKS. 

Ex.  1.     By  a  late  enumeration  of  the  people,  the  num- 
ber of  inhabitants  in  England  is  put  down  at  nine  mil- 
lions, three  hundred  forty  three  thousand,  five  hundred  | 
and  seventy -eight ;  and  the  number  found  to  be  in  Lon- 
don was  eight  huiidnd  eighty-five   thousand,  five  hun-J 
dred  and  eighty -seven  ; — How   are  these  numbers  ex 
pressed  in  figures. 

Ex.2.  'Ihe  world  was  created  two  thousand  three  h 
hundnd  and  forty-eight  years  before  the  Deluge  ;  three ji 
thousand  two  hundred  and  fifty -one  years  before  the 


11 


NUMERATIOJiT,  13 

building  of  Rome  ;  four  thousand  and  four  years  before 
the  birth  of  Christ,  and  five  thousand  and  fourteen  years 
before  the  present  time  [1811]  :— Let  each  of  these  num- 
bers be  expressed  in  figures. 

Ex.  3.     Express  in  words  the  distances  of  the  prima- 
ry planets  from  the  8un,  which  are  as  follow  : 

Mercury  -  -  -  37,000,000     Venus  -  -  -  66,000,000 
'Dhe  Eafth  -  4  95.(^0.000     Mars  -  -  -  l4j,0(-0,r)00 
J^iteijfc.  -%493:b00,%0     Saturn  -    -  903,000,000 
«  w%  u^  Hgscl^el  -  ^1,813.000,000  miles.* 
Fractions,  or  broken  nunibers,  are  expressed  in  the 
following  manner  ;-^-vA  halfperffiy  is  denoted  by  I  ;  a  far- 
thing, by  I,  being  the  one-fourth  of  a  penny  ;  and  three 
farthings  by  |,  being  three-fourths  of  a  penny.      Ihus  it 
appears  th  !t  a  fraction  is  any  part  or  parts  of  a  unit,  and 
is  expressed   by  two  numbers  separated  from  each  other 
by  a  short  line.     The   lower  number  shows   how  many 
parts  the  unit  is  divided  into,  and  the  upper  figure  points 


•  The  ancient  Romans,  in  their  Notation  of  Numbers,  made 
.use  of  tile  tbllowinj^  five  letters  :  I,  V,  X,  L,  and  C,  which 
singly  stood  for  one,  five,  ten,  fifty,  and  a  hundred.  By  re- 
peating^ itnd  combining  these,  any  otlier  numbers  were  expres- 
cd  :  thus  II,  signified  f  wo  ;  111,  three;  XX,  twenty,'  CC,  ttoo 
hundred^  and  so  on.  I'he  rules  tor  Koman  No'tation  are  as  fol- 
low : 

1.  The  antiexing  a  letter  of  a  lower  value  to  one  of  a  higher, 
increases  its  vulue,  or  denotes  the  sum  of  both,  as  VI,  signifies 
six  ;  XII,  denotes  twelve  ;  LV,  fifty -five  ;  LXXVI,  seventy-six ; 
CLII,  one  hundred  and  fifty-two. 

2.  The  prefixing  a  letter  of  a  lower  value,  to  one  of  a  higlier, 
shows  tliat  the  viiiue  of  the  less  is  to  be  taken  from  the  greater, 
ov  sliows  their  dift'trence:  thus,  1  prefixed  to  V,  or  IV,  is  four; 
IX,  nine  ;    XL,  forty;  XC,  nuicty,  he. 

For  the  sake  gf  ^abbreviation,  the  Romans  introduced  these 
marks: — I^,  five  hundred  :  do,  a  thousand,  these,  in  process 
of  time,  were  vvritien  I),  M,  so  ihat  now  the  D  signifies  five 
{hundred,  and  ti)e  Mj  a  thousand;  but  in  tlie  titles  of  many  old 
ibooks  we  find  t lie  other  mode  of  Notation.  The  following  table 
will  exhibit  every  thing  necessary  to  be  known  on  this  subject  t 


14 


NUMERATION. 


out  what  number  of  these  parts  are  contained  in  the  frac- 
tion :  thus  I,  when  standing  for  three  farthinos,  shows 
that  a  penny  is  divided  into  four  parts,  the  5  determines 
the  number  of  those  parts,  and  we  call  it  three-fourths  of 
a  penny. 


TABLE. 


1 

li  - 

III   - 

IV,  or  nil 
V 

VI  - 

VII  - 

VIII  - 

IX  - 
X 

XI  - 

XII  - 

XIII  - 

XIV  - 

XV  . 

XVI  - 
XV[I       - 

XVIII  -    . 

XIX  .     - 

XX  -    - 

XXI  - 
XXX    - 
XL     - 
XLI    -      - 
L        -      - 


^2 
IS 
14 
15 
16 
17 
18 
19 
20 
21 
30 
40 
41 
50 


LX        T  y 

LXX      Lm  '^ 

LXXX  'Z  -- 

XC        '^  - 

c     ^    -  ^  '.\ 
CI 

cr-c    - 

Iq,  or  D 

IjC,  or  DC  - 
loCCC,  or  DCCC 


1 1  I  loCCCC,  or  DCCCC,  or  CM 


CIo,  or  M    - 
CI^C,  orMC 
MiM,  11^  - 
l30t,orV_        . 
10  jM  or,  VI  _^ 
l30MMM,or_Vllt 
CCIoo  t,  or  X_^  - 
CClooM,  or  XI 

IdodMM  .    - 

CCCIOOOM    - 


60 
70 
80 
"90 
100 
101 
300 
500 
600 
800 
900 
1000 
1100 

-  2000 
.  5000 
.  6000 

-  8000 
.  10000 

11000 

5000O 

52000 

lOlOOO 


Cl3l3CCC,XI,orM,l)CCC,Xl  1811 


*  The  word  thousand  is  often  expressed  by  a  line  drawn  over 
the  top  of  a  number  :  thuslc  signifies  ten  thousand  and  M  a 
thousand  thousands. 

t  The  annexing  Q  to  the  number  lo,  increases  its  value  ten 
times  ;  thus  lo^'is  5000,  and  Ioo3  is  fifty  thousand. 

t  The  prefixing  C,  and  at  the  same  time  annexing  a  0  to  the 
number  CIC,  makes  i-ts  value  ten  times  greater;  CCIOO  is 
10,000,  and  CCCIo03  is  100,000. 


ADDITICrN.  15 

Inches  are  usually  divided  in  ei*^hths,  or  eight  parts,  in 
each  inch  ;  a/sd  the  fractional  parts  are  thus  expressed  : 
I  means  three-eights.      |  means  tive  eights, 
I  means  seven-eights.     f   means  four-eights,  equal  to 

one  half. 
Sixteenths  are  likewise  in  common  use,  and  we  say, 


^g'  five  sixteenth.        4^  eleven  sixteenths. 


fifteen  sixteenths. 


ADDITION. 


Addition  teaches  the  method  of  finding  the  sum  or 
tot^l  of  several  numbers. 

Rule  (1.)  Place  the  numbers  under  one  another,  so 
that  units  may  stand  under  units,  tens  under  tens,  ^c. 

(2.)  Add  up  the  figures  in  the  row  of  units  :  set 
down  what  remains  above  the  even  tens,  or  vf  nothing  re- 
mains, a  cypher,  and  for  the  tens  carry  as  many  ones  to  the 
next  column. 

(3.)  Jidd  up  the  other  rows  in  the  same  manner,  and 
in  the  last  eolumn  put  down  the  whole  sum  contained 
in  it. 


Ex.  1.     What  is   the  sum  of  3G84,    4863,  365,  29, 
56874,  and  609  ? 

t684, 
&6^* 

29 

56874 

609 


Answer  -  -  -  66424  is  the  sum  total. 

Proof.     Add  the  numbers  together  in  a  contrary  or- 
der beginning  at  the  top  instead  of  the  bottom. 


16 


ADDITION. 

EXAMPLES. 

345 

8776 

78329 

489 

6734 

87.i93, 

204 

5709 

346^0- 

695  ^ 

9564 

594-17 

7M^ 

3:218 

21004 

27 

4507 
38508 

12345 

2491 

293G68   # 

Ex.  1.  1234 

Ex.  2.  5f32 

E:#3.  P14  % 

3102 

q^41 

m-        m 

3415 

3231 

23-43 

2510 

4322 

1232 

3423 

3413 

4113 

4159 

2342 

2000 

3241 

1122 

5111 

2324 

3in 

2322 

4231 

2322 

5555 

5254 

^.fim 

Ex.  4.  4321 

Ex.  5.  6543 

Ex. 

,  6.  1234 

6125 

2123 

5654 

3246 

4565 

3210 

4350 

4321 

1353 

5432 

2345 

2464 

6312 

6666 

3210 

3424 

5432 

4633 

4301 

lOlO 

5544 

— r 

^ 

Ex.  7.   7654 

Ex.  8.  1357 

Ex. 

9.  7777 

3212 

2464 

4343 

3456 

2013 

6424 

7654 

5765 

3767 

3210 

4324 

5106 

1357 

1067 

2007 

6420 

2132 

7213 

5234 

4126 

6644 

ADDITION. 

\2U 

Ex.  11. 

2345 

Ex. 

12.  9898 

5678 

6-89 

7676 

9876 

9988 

4317 

5432 

7766 

2603 

1357 

5544 

4762 

9864 

3322 

^^437 

2024 

2200 

645S 

6809 

7773 

8764 

8765 

6499 

9538 

4321 

5741 

6749 

17 


Ex.  13.  5162   Ex.  14.  7640  Ex.  15.  49325 


4876 

39 

24609 

4008 

5784 

37485 

3079 

4^04 

16004 

1234 

9865 

23S48 

2341 

6543 

32946 

3468 

2871 

329 

Ex.  16.  5432  Ex.  17.  6905  Ex.  18.  49603 


5789 

324 

50792 

1 254 

24 

4052 

5678 

9 

498.59 

9123 

5068 

6  A 

4009 

4981 

78432- 

5746 

5139. 

29764 

Ex.  19.  erus  Ex.20.  93217  Ex.  21.  8543 


UN 


89678 

76213 

39764. 

56789 

34)67 

78912 

22545 

8')002 

34567 

67890 

4.678 

91874 

1 2932 

545 

43ft(}4 

4)764 

67890 

5 '87 1 

853t>5 

4.;  652 

20502 

g* 


i« 


ADDITION. 

:.  22.  12345 

Ex. 

,23. 

12349 

Ex. 

£4. 

99887 

54321 

56789 

44556 

6'854 

48672 

17280 

58108 

'24 

69776 

49328 

51403 

43509 

98765 

46795 

49312 

432U0 

31274 

56418 

87219 

45670 

43004 

Ex.  ^.  764329  Ex.  26.  527648  Ex.  27.  397648 


S97643 

476239 

473465 

249764 

765473 

247396 

354673 

620728 

4789i3 

576894 

437649 

862759 

35^649 

276354 

380475 

476392 

762938 

928/64 

734629 

476849 

387649 

563:93 

327649 

258763 

Ex.  28.  476293  Ex.  29.  587649  Ex.  30.  53''649 


547689 
356.-43 
827649 
536754 
873649 
567937 
645'64 
786492 


326-53 
4736*9 
5673ifr 
478943 
6 '4^59 
S8'.745 
47  3659 
7684y2 


764:>32 
476.S43 
324768 
976439 
267568 
374689 
567834 
743687 


Ex. 


ADDITION 

• 

31.  S27638 

Ex. 

32. 

432999 

4  619.7 

7fl3427 

5S86T4, 

632'-42 

7e^!i27 

763487 

48"634. 

629-64. 

927865 

394276 

73t2i86 

839467 

47^288 

364237 

367495 

- 

648^.76 

19 


MISCELLANEOUS    EXAMPLES    IN    ADDITION. 

Ex.  I.  Add  to^fther  the  following  sums;  98764, 
397652,  876.  459321,  f  I,  80,  and  76942, 

Ex  2.  Add  39:64,  47652,  34291,  225,  48,  764871, 
and  10000  together. 

Ex.  3.  What  is  the  sum  of  thirty-five  thousand  and 
four  ;  five  hundred  and  forty  thousand,  three  hundred 
and  nine ;  four  hundred  and  twenty-seven  ;  fifty  thou- 
sand nine  hundred  and  eighty;  two  millions  and  five; 
and  seven  hundred  and  seventy-seven  ? 

Ex.  4.  When  will  a  child,  horn  in  1806,  be  forty- 
nine  years  old  ? 

Ex.  5.  How  many  days  are  there  in  the  first  eight 
months  of  the  year,  ^hen  it  is  not  leap  year  ? 

Ex.  6.  How  old  is  the  world  this  year,  1808,  sup- 
posing it  was  created  4004  vears  before  the  bnth  of 
Christ  ? 

Ex.  7.  A  person  at  his  death  left  3287/.  to  his  wi- 
dow ?  to  his  eld'st  son  he  bequeathed  5250/  and  to 
each  of  five  other  children,  he  left  a  thousand  pounds 
less  than  to  the  eldest  son  :  he  left  also  to  a  nephew 
105/.,  and  the  same  sum  to  be  divided  auiong  four  dis- 
tant relations  :  How  much  money  did  he  leave  behind 
him  ? 

Ex.  8.  The  lease  of  my  house  was  granted  me  in  the 
year  1793.  for  ninety-nine  years;  when  will  it  expire  ? 

Ex.  9.  How  many  days  v\ill  there  be  between  January 
the  first  and  November  the  20th,  1808,  being  leap  year, 
both  days  inclusive  ? 


20  ADDITION. 

Fx  10.  What  do  the  following;  sums  amount  to,  1268 
+  86l2-f  10018+275+919+8+550099  ? 

Ivx.  11.  How  manv  chapters  are  there  in  the  several 
books  of  the  New  Testament  ? 

Kx  12.  How  many  chapters  are  there  in  the  several 
books  of  the  Old  Testament? 

Kx.  13.  How  manv  chapters  are  there  in  the  Bible, 
which  consists  of  the  Did  and  New  Testaments  ? 

Ex.  14.  In  travelling  from  London  to  Bath  in  a  post- 
chaise,  for  how  many  miles  shall  1  have  to  pay  ?  The 
distance  from  London  to  Hounslow  is  10  miles,  from 
Hounslow  to  Maidenhead  is  16  miles,  from  Maidenhead 
to  Reading:  15  miles,  from  Reading  to  Spleenhamland 
16  miles,  from  Spleenhamland  to  Marlborough  is  19 
miles,  from  Marlborough  to  Chippenham  is  19  miles, 
and  from  <  hippenhao)  to  Bath  is  13  miles. 

Ex.  15.  How  far  is  it  from  London  to  Harwick  ?  To 
Romford  are  1 1  miles,  from  thence  to  Ini>atestone  12 
miles,  from  Ingatestone  to  Chelmsford  6  miles,  from 
Chelmsford  to  Colcliester  are  21  miles,  and  from  Col- 
chester to  Harwick  20  mile^^. 

Ex.  l6.  In  travelling  post  to  Margate  I  pay  a  shilling 
a  mile  :  How  many  shillings  shall  1  have  paid  at  the 
end  of  the  journey  ?  The  distance  from  London  to 
Dartford  is  15  miles,  from  thence  to  Rochester  is  14 
miles,  from  Rochester  to  Sittinghourne  is  11  miles,  from 
Sittinghourne  to  ('anterbury  i»  15  miles,  and  from  Can- 
terbury to  Margate  is  17  miles. 


(  21  ) 


SUBTRACTION. 


By  Subtraction  we  find  the  difference  between  two 
numbers. 

Rule  ( 1 .)  Place  the  lesser  number  under  the  greater, 
so  that  units  may  stand  under  units,  tens  under  tens, 
Sfc.  /  begin  at  the  right  hand,  and  take  each  figure  in 
the  lower  line  from  the  figure  above  it,  and  set  down  the 
remainder. 

(2.)  If  the  figure  in  the  lower  line  be  the  greater,  add 
ten  to  the  upper  one,  and  then  take  the  lower  one  from 
the  sum,  set  down  the  remainder  and  carry  one  to  the 
next  lower  figure,  with  which  proceed  as  before. 

(3.)  When  the  figure  in  the  lower  line  is  equal  to  that 
above  it,  the  difference  is  nothing,  for  which  a  cypher 
must  be  set  down. 

EXAMPLES. 


From  -  -  -  874698 
Take   -  -  -  561436 


Remainder  313262 


765087 
425436 

339651 


762134 

597082 


165052 


Proof.     Add  the  remainder  to  the  last  line,  and  if 
the  sum  be  equal  to  the  first,  the  work  is  right." 


22 


SUBTRAi 

From  -  -  -  658742 
Take  -  -  -  346121 

OTION. 

390076 
184193 

431267 
280795 

Remainder  312621 

205883 

150472 

Proof  -  -  $587  42 

390076 

431267 

EXAMPLES    FOR    PRACTICE. 


Ex.  1.     4867434     2.6789491      3.5876486     4.3390761 
25S4213         5468354  3564214  1478490 


Ex.5.     7052673     6.9276807     7.7231607    8.9104008 
3860749         4859434         5587465         9031618 


Ex.  9.     6734078  10.  i201832  11.  6000342  12.  lOOO^OO 
5943769  4b76543         5999343  99999(> 


Ex.  13.  4002103  14.  3874205  15.  9000123  16.  5301864 
3987654  1796432  8123456  99 


Ex.  17.  7962038  18.91111118  19.  468103520.  8302697 
6498100         80000009  93006        2912934 


Ex.  21.60001234  22.71216003  23.30061217  24.26013032 
49993490       59876543         19996642        19125346 


SUBTRACTION.  f3 

Ex.  25.  '98743205  26-  50257480  27.    49764321 
9^99999      41926321      1587549!? 


Ex.  28.  93816('86  29.  94286730   30.  92370800 
927908     32199739       4812719 


Ex.  31.  42601304  32.  27000019   53.  76253922 
22500894      4102094        344939 


Ex.  34.  33861400  35.  94681039  S6.    6901090 
23713509       3041316       1860018 


Ex.  37.  591040029  38.  271216904  39.  97348098 
490300019      28391767      9290412 


Ex.  40.  974689019  41.  593902742  42.  913062138 
31689247    312003717      44823165 


Ex.  43.  797260839  44.  170909009  45.  99326104 
62310079      24710905     2128 i 299 


24  SUBTRACTION. 

Ex.  46.  19390P09     47.  3.'921090  48.   1115677333 
2109109  1937(;99  38 103475 


MISCELLANEOUS    EXAMPLES    IN    SUBTRACTION. 

Ex.  1.  The  invention  of  gunpowder  was  discovered 
in  the  year  1302  :  How  long  is  it  since  to  the  present 
year,  181 1  ? 

2.  What  is  the  difference  between  thirty-five  thou- 
sand three  hundred  and  nine,  and  nine  thousand  and 
ninety-nine. 

3.  How  much  does  seven  hundred  six  thousand  and 
four,  exceed  fourteen  thousand  nine  hundred  and  thir- 
ty-seven ? 

4.  How  much  does  fifteen  thousand  and  five  want  of 
twenty-three  thousand  ? 

5.  The  art  of  printing  was  discovered  in  the  year  one 
thousand  four  hundred  forty-nine.  How  long  is  it  since 
1808  ? 

6.  Coaches  were  first  used  in  England  in  the  year 
1580  :   How  many  years  is  it  to  1808  P 

6.  Needle  making  was  introduced  into  England  from 
India  in  the  year  1545:  How  many  years  was  that  be- 
fore the  present  king  came  to  his  throne,  which  was  in 
1760. 

8.  Required  the  answers  of  the  three  following  sums  ; 
18045—999;  2059—928;  and  258764 — 49876. 

10.  How  many  more  chapters  are  there  in  the  Old 
Testament  than  in  the  New  ? 


(25  ) 


MULTIPLICATION. 


Multiplication  is  a  short  method  of  Addition,  and 
it  teaches  us  to  tind  what  a  number  will  amount  to,  when 
it  is  repeated  a  certain  number  of  times. 

Rule.  The  number  to  he  multiplied  is  called  the 
J^Iuttiplicand  :  and  the  number  muttiptied  is  called  the 
JUuUijplier,     The  number  found  is  called  the  Product. 

1  '  MULTIPLICATION    TABLE. 


2  times 
or  twice 
i  are   2 
2  4 


6 

8 
10 
12 
14 
16 

20 
22 
24 
8  times 

1  are   8 

2  lo 


times 

are  3 

6 

9 

n 

18 
21 
24 
27 
30 

3C\ 


3 

4 

5 

6 

7 

8 

9 

10 

11 

12 


4  times 

1  are  4 

2  8 


times 
are  5 
10 
15 
2u 
25 
3u 
35 
40 
45 
50 
55 
60 


6  times 
1  are   6 


1    9  times 

10  tin 

les 

Ill  til 

nes 

1  are  9 

lar 

2   10 

lar 

e  11 

1   2         lu 

t    2 

20 

2 

2z 

3         27 

3 

3v 

3 

33 

4         36 

4 

40 

4 

44 

5         45 

5 

50 

5 

55 

6         54 

6 

60 

6 

66 

7         63 

7 

rv> 

7 

7. 

8         72 

8 

80 

8 

8« 

9        81 

9 

90 

9 

99 

10        90 

10 

loo 

10 

110 

11         0^ 

11 

110 

11 

1^1 

12       108 

12 

12. 

12 

132 

7  times 
1  are  7 


12 
1! 
24 
30 
o6 
42 
48 
54 
60 
66 
72 
y^ 
12  times 

1  are  12 

2  24 

3  36 
48 
60 
72 
84 
96 

108 


120 
133 
144 


y 


26  MULTIPLICATION. 

1.     When  the  Multiplier  does  not  exceed  13. 

Rule.  Multiply  every  figure  in  the  multiplicand  from 
ri^ht  to  left,  consider  how  many  tens  there  are  in  each 
product,  the  remaining  units  set  down  under  the  figure 
multiplied,  and  carry  the,  tens  as  so  many  ones  to  the 
next  product.  The  last  product  is  to  be  wholly  set] 
down. 


Ex.  1.  420847 
8 


EXAMPLES. 

Ex.  2.  94564875 
5 


Ex.    3.  3476819 
12 


3366776 


472824375 


41721828 


Thus  in  the  first  exatnple,  I  say  8  times  7  are  56,  in 
which  there  are  five  tens  and  six  over,  I  put  down  the 
;8ix,  and  say  8  times  4  are  32,  adding  the  5  from  the  last 
product,  1  have  37  ;  1  put  down  the  7,  and  carry  the  3  for 
the  three  tens ;  I  then  say  8  times  8  are  64,  and  3  are 
67,  7  and  carry  6  ;  8  times  0  is  0,  but  put  down  the  6 
brought  from  the  last  product;  8  times  2  are  16,  put 
down  the  6,  and  then  8  times  4  are  32,  and  the  one 
brought  forward  are  33,  which  as  being  the  last  pro- 
duct, must  be  set  down. 

EXAMPLES    FOR    PRACTICE. 


Ex.  1.  4653245 
2 


Ex.  2.  8756894 
3 


Ex.  3.  4986587 
4 


Ex.  4.  S39076S   Ex.  5.  705id673   Ex.  6.  9276807 
5  6  7 


Ex.  7.  7231607   Ex.  8.  9134908   Ex.  Q.   6734078 
8  9  10 


MULTIPLICATION,  Si7 

Ex.  10.5201832     Ex.  U.  6393476     Ex.12.  8874025 
11  12  11 


Ex.  13.  83022697    Ex.  14.  5391864     Ex.  15.  4681953 
12  11  12 


Ex.  16.  9874.3205  Ex.  17.  50947496  Ex.  18.  49764329 
9  8  7 


Ex.   19.  5972834  Ex.  20.  5097648  Ex.  21.  5875496 
5  6  4f 


Ex.  22.  5439027  Ex.  23.  9999999  Ex.  24.  8888888 
7  8  7 


Ex.  25.  9734895  Ex.  26.    9237085  Ex.  27.  5942867 
9  8*  9 


This  character  Xi  wliich  is  cal]«^d  St.  Andrew's  cross, 
is  us'mI  to  denote  Multiplication,  and  when  it  stands  be- 
tween two  numbers,  it  signifies  that  those  numbers  are 
to  be  m'lltinl'ed  into  one  another:  thus  9  x  6  "Z:  54,  is 
re«d,  nine  niuUip!  od  by  six  "s  equal  to  fifty-four.  Asrain 
12  '  11  —132,  that  is  12  multiplied  by  U  is  equal 
te  132. 


28 


i 

MULTIPLICATION. 

EXAMPLES. 

Ex. 

1. 

528318769  X  5 

Ex.  2.  9567283 14  X 

S 

Ex. 

3. 

825934685  X  7 

Ex.  4.486875294  x 

9 

Ex. 

5. 

496745 8 3S  X  9 

Ex.  6.  683637544  X 

8 

Ex. 

7. 

578940245  X  2 

Ex.  8-759654318  X  11 

Ex. 

9. 

987234617  X  6 

Ex.  10.  867122436  x 

12 

Ex. 

11. 

716432978  X  9 

Ex.  12.  6876493^1  X 

7 

Ex. 

13. 

795483206  X  11 

Ex.  14.779368245  X 

9 

Ex. 

15. 

91872648  x  12 

Ex.  16.  986G49005  x 

5 

Ex. 

17. 

85678654  X  4 

Ex.  18.390057864  X 

6 

Ex. 

19. 

894.367542  X  8 

Ex.  20.  765438958  X 

4 

II.  To  multiply  by  10,  add  an  0  to  the  multiplicand  : 
thus  567  X  10  is  5670  ;  and  567  x  100  is  56700  ;  and 
6489  X  1 0000  =  64890000.  Therefore,  to  multiply  a 
given  number  of  one  denomination,  by  a  number  whose 
significant  figures  do  not  exceed  12,  having  a  cypher  or 
cyphers  joined  to  it : 

Rule.  Write  down  the  cypher  or  cyphers  for  the 
first  part  of  the  product  towards  the  right  hand,  and 
then  multiply  every  figure  in  the  multiplicand  by  the 
significant  figures  of  the  multiplier,  as  in  the  preceding 
case. 


Thus,    S469456  X  50  ZT  173472800,    and  98765432  X 
8000  IT  7901 23  456000,  for 

3469456  98765432 

50  8000 


173472800 


790123456000 


EXAMPLES. 

Ex.  1. 

6754328  X  70 

Ex.  2. 

987654329  x 

800 

Ex.3. 

8329674  X  1 10 

Ex.4. 

56780943  X 

120 

Ex.  5. 

6470078  X  9000 

Ex.  6. 

9237654  X 

1100 

Ex.7. 

7856423  X  lOOO 

Ex.8. 

7490434  X 

600 

III.  When  the  multiplier  consists  of  several  figures. 


MULTiriCATlON,  ^9 

Rule.  The  multiplicmd  must  be  multiplied  by  each 
fl2;ure  of  the  multiplier  separately  be^inmn<»;  with  the 
rij^ht  hand  tii^ure,  and  the  first  fij^u re  of  every  product 
must  Hta:jd  exactly  under  the  figure  tnultiplied  by.  Add 
these  products  together  for  the  whole  product. 

To  multiply  by  any  number  between  13  and  19  in  one 
line. 

Rule.  Multiply  the  unirN  figure  of  the  multiplicand, 
by  the  right-hand  «ligit  of  the  multiplier  ;  set  down  the 
uint's  fiiiure  of  the  product,  and  remember  what  is  to  be 
carried.  Multiply  the  second  figure  of  the  multiplicand  ; 
to  the  product,  idd  what  was  to  be  carried,  and  also  the 
first  fiiture  of  t  »e  multiplicand.  Then  set  down  the 
unit's  ti;;ure,  and  retain  in  your  mind  the  number  to  be 
Curried,  as  before.  Multiply  the  third  figure  of  the  mul- 
tiplicand :  add  the  number  to  be  carried,  and  also  the 
second  figure  ol  the  multiplicand,  and  so  on  5  thus 

74365487596 
17 


1C642I3:89132 


Here  I  say  7  times  6  are  42  ;  I  put  down  the  2  and 
carry  4,  and  say  7  times  9  are  63,  apd  4  are  67,  then 
add  the  b,  which  makes  73  ;  put  down  the  3,  and  say  7 
times  5  are  35,  and  7  are  42,  to  which  add  the  9,  which 
make  51,  put  down  1  and  carry  5,  and  so  on,  till  the 
last  figure,  when  I  say  7  times  7  are  49,  and  3  to  be 
carried  are  52,  take  in  the  4,  which  make  56,  put  down 
6,  and  add  7  to  the  5,  and  set  down  \2. 

To  multiply  by  21,  31,  41,  &c.  to  91  in  one  line. 

Rule.  Bringdown  the  unit's  figure  of  the  multipli- 
cand for  the  unit's  figure  of  the  product ;  multiply  the 
same  figure  by  the  left  hand  digit  of  the  multiplier,  to 
which  add  the  next  figure  on  the  left  hand  of  the  multi- 
phcandi  set  down  the  unit's  figure  and  carry  the  tens, 


3f  MULtlPLICATION. 

multiplj  the  next  figure  of  the  multiplicamd  by  the  same 
Multiplier,  and  so  on,  always  observing  to  add  the  num- 
ber you  carry  and  also  the  first  figure  on  the  left  hand  of 
that  which  you  multiply. 

EXAMPLE. 

3760942 
21 


78979782 


Bring  down  the  2  then  say  twice  2  are  4,  and  4-  are  8, 
put  down  8  and  say  twice  4  are  8,  and  9  are  17,  put  down 
7,  and  carry  1 ;  then  say  twice  9  are  18  and  1  are  19,  put 
down  9  and  carry  1  ;  next  twice  0,  will  be  0,  but  the  1 
you  carried,  and  6  make  7,  put  down  7  ;  twice  6  are  12 
and  7  are  19,  put  down  9  and  carry  1 ;  then  say  twice  7 
are  14  and  1  are  15  and  3  are  18  ;  put  down  8  and  car- 
ry 1,  and  lastly  twice  3  are  6,  and  1  are  7. 

EXAMPLES. 


57864329 
579 

35964827 
846 

520778961 
405050303 
289321645 

2157889C2 
143859308 
287718616 

33503446491 

3042624  3642 

Proof.  The  readiest  way  of  proving  the  truth  of 
sums  in  Multiplication  is,  by  casting  out  the  ninesi 

Rule.  Make  a  cross  like  that  which  is  used  to  de- 
note Multiplication  :  add  toj^ether  the  figures  in  the 
•multiplicand,  casting  out  all  the  nines  in  the  sum  as  of- 
ten as  they  amount  to  9,  and  put  the  remainder  down  on 
one  side  of  the  cross  5  do  the  same  with  the  fnultiplier. 


jvruLTiPLiCATiosr.  3X 

and  put  down  the  remainder  on  the  other  side  of  the 
cross  Multiply  the  two  remainders  to}j;ether,  and  cast- 
ing out  the  nines  of  their  product,  will  leave  the  same 
remainder  as  the  nines  cast  out  of  the  answer,  when  the 
work  is  right. 

EXAMPLES. 

459S267  0  7628954         1 

568        OXl  857       5Xt 


0  — I 

56746136  534026:'8 

27559602  38  44*70 

22966335  61031632 


2e.089'^5656  653801^578 


To  prove  the  second  example,  I  saj  7  and  6  are  13  5 

4  ahove  nine,  (omit  the  9)  :  4  and  2  are  6  and  8  are  14; 

5  ahove  nine,  (omit  the  9)  :  5  and  5  are  10,  1  ahove  9, 
1  and  4  are  5  :  I  place  the  5  on  the  left  hand  of  the  cross, 
and  say  8  and  5  are  13,  4  ahove  9  ;  4  and  7  are  11,2 
ahove  9  :  the  2  I  put  on  the  right  hand  of  the  cioss; 
Kovv  5Xfi  gives  10,  which  is  1  above  9,  I  put  the  1  at 
the  top  of  the  cross,  and  then  cast  out  the  9's  of  the 
whole  product,  and  I  find  the  remainder  is  1,  which  an- 
swering to  the  1  at  the  top  of  the  cross,  leads  me  to  con- 
clude that  the  operation  is  right. 

IV.     When  cyphers  are  intermixed  with  the  figures 
In  the  multiplier. 

Rule.     Omit  the  cyphers,  and  let  the  first  figure  of 
each  product  be  placed  under  its  multiplier* 


32  MULTll^XieATlOW. 


EXAMPLES. 


Ex.  1.  76503^.9  Ex.  2.  4465348 

6'i0j09  I                      7()0<  608         3 

5X2 7  X  8 

6P^<5  961  1                     357  -784    .    8 

3S-i5i645  26:9:^0B8 

459' l'r4  31257436 


45^4091417461  31 2601,1093  i  584 


Ex.  3   849275   X  706  Ex.  4.  973648  X  8005 

Ex.  5.  59:384   X   830004     Ex.  6.  364*59    X   2709 
Ex.  7.  245918   X   70.S006     Ex.  8.  609483    X  95007 

V.  When  the  multiplier  is  the  product  of  two  or 
more  numbers  in  the  table. 

Rule.  Multiply  the  multiplicand  by  one  of  the  com- 
ponent parts,  and  that  product  by  ti»e  other,  and  so  or? : 
thus  if  I  have  to  multiply  a  triven  sum  by  64,  I  find 
8  X  8  =  64;  instead,  therefore,  ofmultiplyinwhy  6  and 
4  in  the  usual  way,  I  multiply  first  by  8,  aiid  then  that 
product  by  8  again. 


864392 
8. 

X 

64 

5 
5  X 

5 

EXAMPLES. 

39746285  - 

7 

X  168 

3 

8X6 

S 

6915135 
8 

I  .              278223995 
6 

55321088 

1669343970 
4 

667?  375880 

t 

i 


MULTIPLICATION.  33 

EXAMPLES  IN    ALL  THE  CAS£:S. 


:.  1. 

99365497  X  13 

2. 

54962874  X  26 

3. 

36729876  X  56 

4. 

47893062  X  48 

5. 

73167482  X  77 

6. 

8274386  X  96 

7. 

39745371  X  86 

8. 

5487962  X  357 

9, 

72983456  X  99 

10. 

3891307  X  464 

11. 

737394  X  4567 

]2. 

35846  X  4682 

13. 

S29357  X  2839 

14.. 

,   58427  X  3957 

15. 

462875  X  «874 

16. 

47683  X  3456 

17. 

594326  X  5936 

18. 

87491  X  7892 

19. 

486752  X  4608 

20. 

29687  X  3579 

21. 

8739690279  X  3978i^9 

32. 

7936820056  X  500634 

23. 

25764.'>2874  X  613487 

24. 

9167  4032^8  X  653000 

25. 

872694325  :><  2900008 

26 

715970032  X  350706 

27. 

52673'it69  X  590734 

28. 

37  M5687  X  999999 

29. 

74714323  X  3^5627 

30. 

46382719  x  50000092 

MISCELLANEOUS  EXAMPLES. 


Ex.  1.  ^^ultiply  three   millions   thirty-nine  thousand 
and  three,  bj  thirty-five  thousand  and  twenty-eight. 

2.  Multiply  six  billions,    six  hundred  thousand  and 
sixty-five,  by  eight  thousand  and  thirty-nine. 

3.  There  are  eleven  hundred  hackney  coaches  in  Lon- 
don }  suppose,  on  the  average,  esich  coach  earns  thir- 


34  DlYlSIQNk 

teen  shillings  adaj^  how  many  shillings  will  be  expend- 
ed in  the  hire  of  these  carriages  in  a  year  of  365  days, 
Sundays  being  excepted  ? 

4.  In  Jamaica  only  there  were  imported,  annually, 
not  less  than  ten  thousand  eight  hundred  negroes  from 
the  coast  of  Africa:  How  many  slaves  had  free-born 
Englishmen  made  in  that  island,  between  the  year  1799 
and  the  year  1807,  in  which  the  infamous  traflick  was 
abolished. 

5.  A  boy  can  point  sixtee»thousand  pins  in  an  hour; 
How  many  will  he  do  in  six  days,  supposing  he  works 
eleven  clear  hours  in  a  day?  See  Blair*s  Universal 
Free  ptor. 

f>.  What  is  the  continual  product  of  f5,  19,  703,  and 
999  ? 

7.  How  many  changes  can  be  rung  on  twelve  bells  ? 

8.  Multiply  the  difference  between  50487  and  30056, 
by  the  sum  of  850.  9067,  and  800  ? 

9.  The  sum  of  two  numbers  is  30 ;55,  and  the  greater 
nunjber  is  25S51  ;     What  is  their  product  ? 

10  The  sum  of  two  numbers  is  45 o 4,  and  the  less  is 
1876;  What  is  their  product  ? 

11.  What  is  the  difference  between  twelve  times  fifty- 
seven,  and  twelve  times  seven  and  fifty  ? 

12-  How  many  miles  will  a  person  walk  in  sixty-six 
years,  supposinij  he  travels,  one  day  with  another,  six 
mil*--,  and  there  are  .^65  liays  in  a  year  ? 

13  How  maov  cn;  c  foet  does  this  room  contain, 
which  i>  fifteen  feet  lonj^,  fourteen  feet  wide,  and  thir- 
teen feet  high  ? 


DIVISION, 


Bt  Division,  we  find  how  often  one  number  is  con- 
taineil  in  another  of  the  same  denoiuination ;  this  is  a 
short  method  of  performing  Subtraction. 


»ivisroN.  35 

The  sum  to  be  divided  is  called  the  dividend  ;  the  fi- 
gure, or  figures  by  wliich  we  divide,  is  called  the  divisor  ; 
and  the  result  is  called  the  quotient. 

In  this  Rule,  as  ia  Multiplication,  there  are  several 
distinct  cases. 

I.   When  the  divisor  does  not  exceed  1 2. 

Rule.  Write  the  divisor  on  the  left  hand  side  of  the 
dividend,  make  a  curve,  and  consider  how  often  the  di- 
visor is  contained  in  the  iirst  figure,  or  in  the  first  two 
or  three  fia;nres,  and  set  tlie  quotient  under  it ;  and  for 
every  unit  remaining  after  subtraction,  carry  ten  to 
the  next  figure  of  the  dividend. 

EXAMPLES. 

Ex.1.     4)78654328  Ex.2.     9)85674327 


•19663582  95iy3o9— 6 


Ex.3.     11)10876341  Ex.4.     12)11272459 


988776—5  939371—7 


In  the  second  example,  I  saj  there  are  9  nines  in  85 
and  4  over  ;  I  put  down  the  nine  and  carry  the  4,  as 
40  to  the  6,  and  the  9*s  in  the  46.  ,5  times  and  1  over  5 
put  down  the  5  and  carry  1,  as  JO.  and  say  the  9's  in 
17,  once  and  8  over;  put  down  the  1  and  carry  8 
as80  j  9's  in  83,  9  times  and  two  over,  and  so  on  :  at>* 
the  last  figure  there  are  6  remaining,  put  down  this  be- 
yond a  small  line. 

It  is  usual,  in  giving  the  answer,  to  make  a  short  line 
under  the  remainder,  and  place  under  it  the  divisor  ; 
tlius  the  answer  to  the  secori<l  sum  is  9519369|^ :  that 
of  the  third  sum  is  988776^^5^.,  and  that  of  the  fourth 
939371  ^z  ;  and  the  three  remainders  are  fractions, 
which  we  read  six-ninths,  five-eleventhsj  and  seven- 
twelfths,    bee  p.  4.  and  5. 


$6 


DIVISION". 


5)763948r 


15278971 


EXAMPLES. 

7)440295 
6289yf 


8)5678943 
7098671 


This  character  h-,  when  placed  between  two  numbers^ 
signifies  that  the  one  is  divided  by  the  other ;  thus 
95  -T-  8  =  1 1|  ;  and  we  read  93  divided  by  8,  ^ives  1 1 
and  seven-eights  over  ;  that  i^,  there  are  eleven  eights  m 
95,  and  seven  remaining. 


EXAMPLES. 


Ex.  I.  5687  —  7 

Ex.  2.  49876  -r-  3 

Here  5687  -h  7  =  812|-. 

4i;8*6  ^  3  =  166254 

For  7)3687 

3)49876 

812—3 

J  66.^5— -1 

Ex.  3.  87240322  -J-  3 

Ex.  4.  62304678  -i-    4 

5.  74009634  -^  5 

6.  .6'30%ir  ~  6 

7.  59-34600  -T-  7 

8.  37026;41  ^  9 

9.  46872135  -~    8 

10.  3643875J  —    7 

11.  439^0361  —  9 

12.  3256487  -^    8 

13  51'764218-r- 10 

14.  3.>3'2640  -T-  11 

15.  327-3742  -t-  12 

16.  333u333  ~  12 

17.  44444444  -—  H 

18.  5598764  -^  12 

19.  98897bU3  -~    9 

20.  933U048  -r-  8 

Proof — I'lie  method  of 
Division,  IS  to  multiply  the 
take  in  the  remainder,  the 
dividend. 


proving  the  truth  of  sums  in 
answer  by  the  divisor,  and 
result  will  be   equal  to  the 


Ex.     7959467b'?4  ~  7 
7)7959467834 


Quotient-  1137066833—3 

7 


8 
7X2 

8 


Proof  -  -    7959467834 


WIVISION.  Sf 

Another  method  is  bj  casting  out  the  nines,  as  in 
Multiplication. — Rule.  Cast  away  the  nines  in  the 
divisor,  and  put  the  remainder  on  one  side  of  the  cross  ; 
then  for  the  top  figure  multiply  these  two  numbers  to- 
gether, cast  away  the  nines,  and  add  the  excess  of  nines 
in  tJie  remainder  after  division,  and  the  excess  of  nines 
in  this  sum  will  be  equal  to  the  excess  of  nines  in  the 
dividend,  if  the  work  is  right.  See  the  preceding  exam- 
pie,  where  I  put  down  the  7  on  one  side  of  the  cross  ;  do 
the  same  with  the  quotient,  for  the  other  side  of  the  cross  : 
the  excess  of  nines  in  the  quotient  is  2,  which  I  put  ow 
the  other  side  of  the  cross,  then  I  say  7  times  2  are  14, 
and  the  remainder  3  make  17,  which  is  8  above  nine, 
this  I  put  at  the  top  of  the  cross,  and  T  find  that  8  is  the 
excess  above  the  nines  in  the  dividend,  therefore  I  con- 
elude  the  operation  is  right. 

II.  To  divide  a  number  of  one  denomination,  by 
another  number  whose  significant  figures  do  not  exceed 
12,  having  a  cypher  or  cyphers  joined  to  the  right  hand. 

Rule.  Cut  off  the  cyphers  from  the  divisor,  and  the 
same  number  of  figures  from  the  right-hand  of  the  divi-. 
dend  ;  then  divide  the  remaining  figures  of  the  dividend 
by  the  remaining  part  of  the  divisor,  and  the  result  is 
the  anstver. 

To  the  remainder,  if  any,  join  those  figures  of  the 
dividend,  which  were  first  cut  off,  and  the  whole  will  be 
the  true  remainder. 

Divide  46R5321   by  800  ;  and  326441  by  1200. 
8.00)46853  21  12.00)3264.41 


^856—521  2:2—41 

Of  course  the  true  answers  to  these  sums  are  5856|U, 
an4  272^1^ 


S8 

BIVISION, 

EXAMPLES 

Ex.  1. 

3476521  -7- 

60 

Ex. 

2. 

8543009  -^  700 

■  3. 

2y.'i7()48  -f- 

8o0 

4. 

9C034  6  -T-  9000 

5. 

5620042  -T- 

noo 

6. 

7641121  -^  500 

7. 

40  2079  -^ 

1200 

8. 

84^6531  -r-  1»000 

9. 

7021164  -f- 

90 

10. 

993' 216 -r-  80f)0 

11. 

46201 132 -i- 

700 

12. 

1234567  -r-  120 

IIT.  To  divide  a  given  number  of  one  denomination, 
by  a  divifior  which  is  compounded  of  two  or  more  num- 
bers in  the  Multiplication  Table. 

Rule.  Divide  the  given  number  bj  one  of  those 
parts,  and  th^*  quotient  by  the  other  component  part,  and 
so  on  till  each  of  the  component  parts  has  been  used  as  a 
divisor;  thus  46875815777^  105  is  performed  as  fol- 
lows :  the  divisor  105  is  equal  to  7  x  5  X  3  ;  I  there- 
fon  divide  the  dividend  first  by  7,  and  the  quotient  by  5, 
and  this  second  quotient  by  3. 


Ex 


7)fl6875815777 

5)6696545111 

3); 339309022. 

-.  r 

I 

Answer  -  -  446436340  • 

--2_ 

\ 

examples 

• 

.  1. 

84596543  -r-  36    Ex.  2. 

545069549  -r- 

42 

3. 

45897642  -r-  56 

4. 

94^960542  — 

99 

5. 

39200761  -T-66 

6. 

87932874  -r- 

768 

7. 

38426587  -r-  550 

8. 

44444444-7- 

121 

9. 

28476974  -^  720 

10. 

55555555  H- 

37S 

11. 

56342b72  -J-  132 

12. 

33992288  -f- 

288 

13. 

34765982 -T-  144 

14. 

9845S392  -r- 

432 

35. 

24853274  -f  512 

16. 

83547552  -r- 

99 

17. 

43.-33999  -h  343 

18. 

54954535  -^ 

720 

19. 

5555556  -^  729 

20. 

25574538  -r- 

343 

DIVISION.  3i> 

IV.  To  divide  by  a  number  cansistinj^  of  two  or 
more  dij^its,  which  Dumber  is  not  compounded  of  those  in 
the  table. 

Rule  (I.)  Draw  a  curved  line  on  the  right  and  left  of 
the  dividend,  and  write  the  divisor  on  the  left. 

(2.)  Find  how  many  times  the  divisor  is  contained  in 
as  many  figures  of  the  dividend  as  are  just  necessary, 
and  place  the  number  on  the  right  for  a  quotient. 

(3.)  Multiply  the  divisor  by  the  quotient  figure,  and 
place  the  product  under  the  above-mentioned  figures  of 
the  dividend,  subtract  this  product  from  that  part  of  tha 
dividend  under  which  it  stands,  and  bring  down  the 
next  figure  in  the  dividend,  or  more  if  necessary,  to  the 
right  hand  of  the  remainder,  and  proceed  as  before,  till 
the  whole  is  finished.     This  is  called  Long  Division. 

Ex.  5537049  H-  954 
954)5537049(5804  Quotient. 
4770... 


7670 
7632 


3849 
3816 


•      33  Remainder.     Answer  5804-^^j\, 

Here  the  divisor  not  being  contained  in  the  first  three 
figures,  I  consider  how  often  it  is  contained  in  the  first 
four,  and  find  it  to  be  5  times,  the  5  I  put  in  the  quotient, 
and  multiply  the  divisor  by  it,  setting  the  product  under 
the  dividend.  I  now  subtract  this  product,  and  to  the 
remainder  767, 1  bring  down  the  0,  and  find  that  the  di- 
visor is  contained  8  times  in  7670,  the  8  I  place  in  the 
quotient,  and  proceed  to  multiply  the  divisor  by  it ;  the 
product  subtractetl  leaves  only  38.;  I  now  bring  down 
the  4,  but  the  divisor  not  being  contained  in  384,  I  put 
down  0  in  the  quotient,  and  bring  down  the  9,  the 
remaining  figure  in  the  dividend,  and  proceed  ^s  before 


40  DIVISION. 

EXAMPLES. 

Ex.  1.  78654321  -J-  76    Ex.  2.  56943278  -~  97' 

3.  68742164  -i-  87  4.  84365487  ~  69 

5.  77755502  -J-  654  6.  45687403  h-  187 

7.  53-430432  ~  7654  8.  56943286  -r-  429 

9.  57078443  -r-  8439  10.  58456942  -^  327? 

11.564320376-7-3976  12.  92876487  -i- 7392 

13,677744032-^-5189  14.  468592 10  -j-  1437 

15.627432871-7-4967  16.  35555555-7-7777 

17.  44444444  -r-  5555  18.  888000999  -r-  999 

19.  33333333-7-999  20.111111111-7-7777 

Ex.  21.  487264325876  ~-  56780909 

22.  87(3842'>8762J  -t-  9095;.843 

23.  948318296542-4-56400032 

24.  5678432 '•i549  -r-  64785321 

25.  877896543210  -i-  92836058 

26.  444444*44444  -■- 750000564 

27.  222000333^046  -7-  708385032 

28.  5409»i53  :876i  -7-  5406057 

29.  32899438654  -^  10010433 

30.  784363254871  -7-  99834369 

MISCELLANEOUS  EXAMPLES. 

Ex.  1.  Divide  fifty  millions  by  four  thousand  and 
seventy -nine. 

2.  The  planet  Mercury  goes  round  the  sun  in  88 
days,  which  is  the  length  of  her  year,  how  many  years 
of  Mercury  wauhl  make  50  of  our  years,  supposing  each 
year  contained  exactly  365  days  ? 

3.  It  is  estimated  that  there  are  a  thousand  millions 
of  inhabitants  in  the  known  world  :  if  one  thirty-third  of 
this  number  die  annually,  how  many  deaths  are  there  in 
a  year  ? 

4.  The  national,  debt  at  present,  cannot  be  less  than 
five  hundred  millions  sterling  :  how  long  would  that  be 
in  piyiii.1  off,  at  the  rate  of  two  millions  and  twenty-five 
pounds  per  annum  ? 


DIVISION.  41 

5.  The  taxes  annually  collected  amount  to  full  thir- 
ty-three millions  of  pounds  :  how  many  poor  families  of 
six  persons  each  would  that  sum  supporl,  supposing  the 
annual  expenses  of  the  father  and  mother  to  be  20/.,  and 
of  each  child  7l.  ? 

6.  My  friend  is  to  set  sail  to  Jamaica  on  the  first  of 
March,  1812,  the  distance  is  reckoned  to  be  3984  miles 
from  England,  at  what  rate  will  he  go,  supposing  he 
reaches  the  Island  on  the  10th  day  of  April,  that  is,  in 
41  days  ? 

7.  What  is  the  diiference  between  the  12th  part  of 
20,100  and  the  5th  part  of  9110  ? 

8.  The  prize  of  30,000/.  of  the  last  Lottery  became 
the  property  of  15  persons  :  how  much  was  each  per- 
son's share,  after  they  had  allowed  750/.  to  the  office- 
keeper  for  prompt  payment  ? 

9.  The  sum  of  two  numbers  is  1440,  the  lesser  is  48 : 
what  is  their  difference,  product,  and  quotient  ? 

10.  The  crew  of  a  ship,  amounting  to  124  men, 
have  to  receive,  as  prize-money,  1890/.  ;  but  as  they 
are' to  be  paid  oft',  they  determined  to  make  their  com- 
mander and  boatswain  a  present,  the  one  of  a  piece  of 
plate,  value  25/.  ;  the  other  of  a  whistle,  which  is  to 
cost  5/. :  how  much  will  each  receive  after  these  deduc- 
tions are  made  ? 

11.  In  all  parts  of  the  world  a  cubical  foot  of  water 
weighs  1000  ounces:  how  many  pounds  are  there,  sup- 
posing 16  ounces  make  a  pound  ? 

12.  A  cubical  foot  of  air  weighs  one  ounce  and  a 
quarter,  how  many  pounds  avoirdupois  of  air  does  a 
room  contain,  which  is  10  feet  high,  14  feet  wide,  and 
16  feet  long? 

13.  Hydrogen  gas,  or,  as  it  was  formerly  called,  in- 
flammable air,  that  is,  the  gas  vvith  which  balloons  art 
filled,  is  full  nine  times  lighter  than  the  common  air 
which  we  breathe  :  how  much  less  would  a  balloon,  con- 
taining 27,000  cubical  feet,  weigh  if  tilled  with  hydrogen 
gas,  than  if  filled  with  common  air  ? 

14.  At  what  rate  per  hour  and  per  minute  does  a 
place  on  the  equator  move,  supposing  the  great  circle 
of  the  earth  to  be  25,000  miles,  and  the  earth  to  turn  on 
its  axis  exactly  in  24  hours  ^ 

4* 


(  42) 


COMPOUND  ADDITION, 


ADDITION  OF  MONEY. 


PENCE    AND    SHILLING     fABLES. 


Pence 

s. 

d. 

Pence 

s. 

d. 

Shill. 

L.  s. 

d: 

20 

-  are 

1 

8 

12  are 

1 

0 

20 

1   0 

0 

«5 

2 

I 

18 

1 

6 

25 

1   5 

0 

30 

2 

6 

24 

2 

0 

30 

1  10 

0 

35 

2 

11 

30 

2 

6 

35 

1  15 

0 

40 

3 

4 

36 

3 

0 

40 

2     0 

0 

45 

3 

9 

42 

3 

6 

.^0 

2   10 

0 

,50 

4 

2 

48 

4 

0 

60 

3  0 

0 

55 

4 

7 

34 

4 

6 

70 

3  10. 

0 

60 

5 

0 

60 

5 

0 

80 

4  0 

0 

65 

5 

5 

66 

5 

6 

90 

4  10 

0 

70 

5 

10 

72 

6 

0 

100 

5  0 

0 

75 

6 

3 

78 

.  6 

6 

110 

5  10 

0 

80 

6 

8 

84 

7 

0 

120 

6  0 

0 

85 

7 

1 

90 

7 

6 

130 

6  10 

0 

90 

7 

6 

96 

8 

0 

110 

7  0 

0 

95 

7 

11 

102 

b 

6 

150 

7  10 

0 

100 

- 

8 

4 

108 

9 

0 

|60 

8  0 

0 

i05 

. 

8 

9 

114 

9 

6 

170 

8  10 

0 

110 

. 

9 

2 

120 

10 

0 

180 

9  0 

0 

115 

. 

9 

7 

1S2 

1 1 

0 

(90 

- 

9  10 

0 

150 

- 

JO 

0 

144 

12 

0 

20O 

- 

10  0 

0 

UNITED  STATES,   OR  FEDERAL  MONEY^ 

10  Mills  (m.)  make   1  Cent,  c. 

10  Cents        —  1  Dime,  d. 

10  Dimes       — —    1   Dollar,  D.  or  g 

10  Dollars 1   Ka-le,  E. 

lOOCts.— — »1 


COMPOUND    ADDITION.  4»5 


ENGI/ISH     MONEY* 


4.  Farthings  (qrs.)  make  1   Penny,  d. 

12  Pence  I  Shilling,  s. 

20  Shillings 1  Pound,  £, 


Compound  Addition  is  a  method  of  collecting  seve- 
ral numbers  of  the  different  denominations  into  one  sum. 

Rule  (1.)  Arrange  the  numbers  so  that  those  of  the 
same  denomination  may  stand  directly  under  each  otheri 
and  draw  a  line  under  them. 

(2  J  Add  the  numbers  in  the  lowest  denomination  to- 
gether, and  find  how  many  units  of  the  next  higher  de- 
nomination are  contained  in  their  sum. 

f3.J  Write  down  the  remainder,  and  carry  the  units 
to  the  next  higher  denomination,  and  proceed  so  to  the 
end. 

L,     s.    d,  I  first  add  together   the  farthings, 

Ex,  4.68   19     Al  which  I  find  to  be  14,  but  14  farthings 

J 23  J 6   !l4  make  2>ld»    I  put  down  the  3  and  carry 

987   12    9  the  3  to  the  column  of  pence,  which  I 

654  i3     7l  then  add  together,  and  find  the  sum  to 

123  17    4a  be  58,  but  by  the  table,  55  pence  are 

45t>  18   \0\  4s.  7rf.,  therefore  58  pence  are  4i».  lorf., 

439     4     Oa  1  put  down   the  10  and    carry  the  4 

592  12    4j  to  the  column  of  shillings  ;  i  now  add 

■  the  •'hillings  together,  and  find  the  sum 

3847    15    lU^  to  be  115,  but  U5   shillings   make  5l, 

^  15s.,  1  put  down  the  15^  and  cary  the 

5  to  the  pounds,  and  proce«d  as  in  simple  addition. 

EXAMPLES    OF    money. 

L.  s.  d.  L.  s.  d,  L,  s.  d.  L.    s.  d. 

Ex.  1.  55  3  8  2.  67  2  8  3.  95  2  9   4.  49  9  II 

62  6  3  24  9  9  89  7  8  33  8  7 

90  2  1  38  2  5  72  4  3  96  12  9 

31  8  4  42  5  9  67  9  2  75  3  4 

43  7  5  78  6  6  51  8  9  51  8  9 

10  9  8  ti4  d  9  45  5  4  12  19  7 


44  COMPOUND    ADDITION. 

X.        S.       d,  L.         S.      d, 

Ex.    5. 


58 

15  9 

6.  42 

16  9 

7.  92 

13  41 

79 

5  5 

37 

15  11 

84 

14  9 

61 

7  10 

73 

9  9 

73 

18  4* 

64 

16  3 

62 

10  6 

69 

17  10 

32 

15  10 

29 

4  4 

48 

15  7 

19 

12  8 

19 

17  11 

35 

14  11* 

L.     s,      d:  L.     s.     d,  L.     s.     d. 

Ex.  8.  50     19     8l       9.54     17     6|     10.67      16     8^ 


97   16 

n 

93 

12 

8 

71 

13  9 

35   14 

2 

31 

6 

9i 

84 

.   11   81 

46  16 

8^. 

25 

10 

11 

32  19  3 

07      16 

2 

76 

13 

10 

48  10  4i 

24   15 

9\ 

44 

6 

61 

55  18  7i 

___ 

33 

8 

3 

21 

1   12  4 

L. 

s. 

d. 

L. 

s. 

d. 

Ex.  11. 

18 

14 

81 

12. 

41 

15 

9^ 

93 

15 

10^ 

56 

10 

9 

37 

6 

11 

62 

16 

31 

78 

16 

5l 

87 

4 

69 

12 

y\ 

78 

13 

n 

43 

8 

11 

92 

19 

01 

12 

17 

SI 

13 

16 

7 

B.  cts.  mis 

i. 

D.  cts.  mis. 

" — 

Z>.  cfs.  Wis/. 

L  73  14  5 

14. 

84 

13 

8 

15. 

69  17  4 

27  37  4 

79 

57 

3 

37  16  2 

46  18  3 

99 

14 

7 

48  27  6 

74  29  9 

37 

74 

5 

62  74  3 

38  17  4 

29 

18 

6 

73  65  7 

85  63  7 

47 

13 

2 

18  11  1 

COMPOUND    ADDITIOJJ.  45 


Ea,     D. 

rf. 

c. 

»n. 

Ea. 

n. 

d. 

c.  w. 

16.  34   4 

7 

6 

3 

17.174 

3 

4 

2  4 

29   3 

2 

7 

6 

27 

4 

2 

6  3 

13   4 

1 

0 

2 

149 

7 

3 

2  8 

ir  6 

0 

2 

7 

76 

4 

2 

9  7 

39   4 

2 

1 

8 

37 

5 

6 

4  7  ■ 

4S   9 

1 

2 

7 

59 

7 

4 

2  6 

L.  s. 

d 

Z. 

s.    d. 

X. 

s.   rf. 

18.  46  2 

H 

19.  45 

19  94 

20. 

43 

17  10| 

65  10 

H 

63 

17  111 

50 

14  6* 

6  4 

74  0 

10 

79 

13  51 

72 

81  17 

H 

46 

10  9| 

65 

19  7», 

39  15 

10 

35 

8  7 

91 

5  1^ 

23  10 

8| 

47 

19  10} 

38 

19  10 

19  14 

^i 

19 

14  6 

29 

12  9^* 

L.   s. 

e£. 

L. 

s,    d. 

L. 

s,      d. 

«1.  52  18 

10 

22.  77 

15  4\ 

23. 

,  57 

15  91 

67  12 

2i 

69 

10  9i 

64 

9  2 

77  14 

9 

41 

0  lOi 

76 

17  \0l 

82  13 

JO^ 

57 

13  8 

97 

IS  9 

98  12 

11^ 

87 

9  lOi 

39 

18  111 

21  17 

■71 

91 

16  U| 

45 

10  10 

45  12 

9 

76 

i4  8 

59 

17  9 

L.  s. 

rf. 

L. 

s.  d. 

L. 

s.     d. 

24.  446  19j^l 

25.   48 

14  10^ 

26 

;.  92 

19  91 

152  1^ 

^10| 

36 

13  10 

56 

10  9 

695  1-2 

0^ 

74 

15  71 

64 

18  71 

7o8  3 

5 

23 

18  2i 

38 

16  3 

S3H  14 

3^ 

48 

9  6 

49 

15  lU 

166  19 

11 

81 

16  4| 

64 

19  SJ 

279  12 

91 

77 

'11  M 

92 

ir  8^ 

46  COMPOUND  ADDITION. 


L.   s.    d. 

L.    s. 

d. 

L.  s. 

A 

27.  1^2  14     91 

28.  54  11 

10 

29.  414   19 

9 

93  16  10^ 

22  19 

&\ 

627    17 

111 

17  12  U 

61    16 

% 

741      f) 

41 

56  13     7\ 

14  17 

01 

865   14 

8 

91    19  11 

58  12 

11.^ 

917     6 

101 

76    14     5\ 

72   10 

6 

347  14 

101 

14    11     3 

r^i   14 

11 

44^    13 

4 

30.  427    18   10* 

31.  548   11 

6 

32.  493     2 

SI 

941   17     9 

932  18 

41 

347  14 

31 

712  19     6 

379     0 

61 

729  19 

5 

6:25   12     7  J 

414   17 

0^ 

672     5 

85 

511    11   10 

573     4 

31 

548   10 

3 

46^2  10     6^ 

697  13 

% 

217   12 

SI 

383   11     9^ 

551     6 

11 

974  16 

7\ 

146     5 

OJ 

33.  412     9  Hi 

34.  152  15 

21 

35.  504     3 

92 

924  19     65 

255  18 

61 

636  15 

5 

750  11     35 

348   12 

% 

421     2 

71 

627    IQ      na 

Atf\          f\ 

t  rt 

^A7  Iv 

10 
01 

438   10     4i 

566  13 

otf    12s 

383     7 

363     2  105 

631     6 

45 

848  15 

2i 

221    15     8 

781     3 

10 

710     0 

81 

147     1     5 

949  16 

7 

483  10 

45 

123   15 

11 

426  19 

7 

36.  576  14     9 

37.  827    18 

111 

38.  792   19 

31 

613   12  U^ 

550  11 

81 

437    14 

9| 

719   13     41 

938     9 

4 

354   10 

lOl 

914  14     Q\ 

344     0 

3 

516   18 

4 

9,n    10     9 

615   16 

U 

209    13 

105 

759     8     5\ 

471      2 

7 

524    17 

2| 

433  15     35 

214  15 

loi 

739     6 

10 

918   11     4| 

745    19 

2 

365     2 

61 

564     7     2 

90  9 

9 

147    17 

9 

_, 

irOMPOUND    ADDITION.  47 

L,  s.    d.  L»    s.     d.  L.  s.     ,d 


39,  88    16    1 

m 

40.  28 

9 

41 

41.  60  15 

55 

26    11. 

51 

54 

17 

9 

48    13 

|3 
»4 

9     7 

2^ 

6 

0 

11 

93    18 

6 

36   13 

4| 

28 

13 

51 

7     7 

105 

41    18 

3 

65 

18 

71 

35    19 

41 

27     3 

81 

92 

6 

41 

73     6 

91 

54  15  ] 

u^ 

7 

16 

01 

31    17 

3 

12   1  J 

6 

14 

5 

10 

59    14 

1(^ 

20     0 

10 

40 

0 

9 

60     0 

10 

42.  94     1 

9i 

43.   53 

11 

4^ 

44.   68   19 

51 

88     2 

6*a 

6 

2 

8 

84     7 

35 

46     5 

Hi 

18 

5 

31 

8     6 

51 

29    16 

35 

26 

10 

75 

25    11 

95 

48    '2 

0 

42 

0 

41 

9    13 

7 

'5    17 

7 

64 

2 

2 

47   15 

61 

61    13 

3i 

71 

18 

icl 

32      1 

3 

7,  14 

101 

3 

14 

115 

1    18 

05 

12  18 

•^1 

80 

0 

61 

2   16 

4 

~ 

45.    76   12 

8^ 

46.  39 

14 

45   ^ 

47.  78  12 

5 

40     0 

61 

97 

12 

21 

17    14 

85 

8   17 

4 

73 

15 

105 

35     0 

6 

24    19 

51 

6 

10 

111 

28   16 

iOl 

59    15 

21 

30 

2 

9 

11      8 

35 

82     6 

51 

16 

12 

51 

49   15 

71 

7    18 

H 

58 

16 

15 

6   11 

41 

S3     2 

91 

2 

13 

7 

62  15 

8 

8  10 

01 

82 

0 

35 

5  18 

41 

10 

10 

10 

90     0 

10 

48  eOMPOUND    ADDITION, 


L.    s. 

d. 

L,     s.    d. 

L.    s. 

d. 

48.127  10 

\ol 

49.515   14     9| 

50.  657    16 

\^\ 

356   14 

91 

043   17     Si 

734  17 

4^ 

483     9 

< 

623  15   11* 

879   14 

^l 

849     7 

11 

417    19     S* 

919   12 

10« 

680  18 

lU 

338   14  10 

131    19 

11 

774   19 

n 

385   18   \M 

235     7 

6J 

114     6 

2^ 

764  13     6 

496  18 

SI 

251    18 

9^ 

453  19     91 

587     9 

5 

428  15 

6 

5m   18     51 

673  U 

10 

567   16 

2 

223    14     2 

820   19 

4 

.4;n    16 

9 

52.  7tf2  10 

9i 

53.477   16 

41 

272    15 

61 

966     4 

81 

395    15 

^ 

889    17 

10^ 

899    13 

6 

736     5 

11 

647    19 

21 

248    16 

101 

692   14 

91 

398   16 

7 

53*2   14 

9 

565    13 

•51 

563  16 

loi 

476    19 

'^i 

937   17 

0 

770     0 

51 

744    12 

9^ 

441    16^ 

41 

945    17 

7 

6f^9  15 

71 

7ro  m 

01 

420   13 

9i 

593    15 

1 1* 

672   11 

11 

150  10  0       150  10  O        40  0  10 


54.  494274 

12 

91 

765502 

6 

4 

S00089 

2 

21 

402193 

If 

9 

375451 

3 

10 

269440 

18 

61 

123428 

15 

10 

567865 

11 

910649 

10 

6 

55,   901442  16  10| 


23497 1 

5 

91 

567 S 52 

14 

71 

912261 

19 

21 

34551^ 

17 

9S 

678830 

12 

6 

912887 

19 

10 

456713 

10 

31 

891391 

17 

81 

COMPOUND    ADDITION. 


4^ 


L, 

s.  d: 

L.     s.    d 

L.      s.     d. 

5'&,  4567 

14  11^ 

57.  3^256  19  6*  53. 

3567  12  9\ 

4934 

15  9 

4397  10  111 

7960  17  10 

2765 

16  10* 

1974  12  9i 

1234  15  71 

9876 

19  lU 

7-246  8  4 

5678  12  8* 

34'J7 

9  5 

S94-2  15  lOl 

9123  14  10 

1234 

10  8^ 

4567  8  91 

4567  13  lU 

5678 

16  10 

4567  17  111 

8912  17  9 

4376 

8  9 

9376  12  8* 

1456  9  6 

£794 

15  4| 

4623  2  5 

7891  10  41 

7921 

12  lOl 

5932  5  4 

2845  ^  3 

59. 1764 

13  91 

60.  2487  7  3  61. 

,  6789  12  5] 

1805 

17   4 

5764  16  11* 

2345  13  11 

1704. 

12  7 

12:?4  J  8  2 

6789  16  9l 

3459 

15  11 

5678  19  91 

4972  15    10 

2946 

16  10^ 

9012  17  10 

3456  19  5* 

1796 

14  10 

3456  2  2 

7891  16  71 

4325 

16.  8 

7890  14  5 

2345  14  11^ 

5678 

12  111 

1234  13  10 

6782  12  9 

4932 

14  6 

5678  15  7 

4315  11   71 

2005 

9  51 

9123  13  4 

2105  8  6 

50 


COMPOUND    ADDITIOX. 


EXAMPLES  OF    fVEIGHTS  AND  MEASURES. 

TROY   WEIGHT. 

24-  Grains  {s,\\)    mako     1    Penny  wt.  pwt 

20  Penny  wt.  —    1    Ounce,  6z. 

12  Ounces        — 1    Pound,   lb. 

Note. — By   this   weight  are   weighed   Gold,   Silver, 
Je'wels,  Liquois,  &c. 

In  adding  up  the  column  of 
grains  I  find  the  sum  to  be  122, 
which  I  divide  by  24  to  bring  it  into 
pennyweights;  and  122  £,rains 
make  5  pennyweights  and  2  grains 
over  ;  the  2  I  put  down,  and  carry 
the  5  to  the  column  of  penny- 
weights ;  [  then  add  these  togetfier, 
and  find  the  sum  to  be  101,  which 
i  divide  by  20  to  bring  to  ounces,  I 
put  down  the  1  and  carry  5  to  the 
coluiiin  of  ounces  ;  then  adding  the 
ounces,  I  find  the  sum  79.  which,  by  dividing  by  12, 
give  6  lb.  7  oz.  the  7  I  put  down,  and  carry  the  6  to  the 
pounds,  and  proceed  as  in  simple  Addition. 


lb. 

oz. 

dwtj 

^.gr. 

7684 

9 

16 

22 

1234 

11 

5 

19 

9876 

8 

11 

22 

14^)3 

9 

19 

12 

3587 

10 

10 

3 

2345 

7 

6 

15 

678P 

9 

14 

21 

3257 

11 

15 

8 

3627  1 

7 

1 

2 

lb. 

oz. 

dwt. 

lb. 

oz.  dwt. 

S''- 

lb. 

oz. 

dwt. 

i.414 

9 

14 

2. 

410 

9   12 

19 

3.  526 

10 

19 

617 

5 

13 

342 

11  16 

12 

712 

9 

17 

713 

10 

9 

912 

3   14 

14 

944 

6 

14 

322 

7 

15 

751 

6   10 

22 

633 

10 

11 

413 

2 

10 

626 

10  17 

16 

319 

4 

10 

5\4- 

11 

15 

427 

4   11 

23 

247 

9 

12 

976 

8 

7 

- 

123 

11   17 

12 

123 

10 

17 

oz. 

lb. 

oz. 

dwt. 

Rr. 

[■:.      OZ. 

dwt. 

dwt, 

•?;r. 

k  9'R) 

10 

]9 

15 

5. 

174  11 

19 

6.  174 

19 

23 

738 

6 

4 

23 

7  4  10 

13 

714 

11 

14 

6J4 

3 

17 

13 

944.  9 

14 

714 

0 

19 

546 

7 

16 

19 

7  1  11 

19 

74 

1 

22 

321 

10 

5 

22 

944  10 

13 

948 

o 

21 

230 

9 

15 

15 

74  11 

3 

74 

2 

12 

946 

11 

19 

i23 

12  4 

6 

301 

14 

4 

COMPOUND    ADDITION.  51 


lb. 

oz. 

(Iwt 

oz. 

dwt. 

s;r. 

7.71 

11 

19 

8.74 

19 

23 

dl. 

8 

14 

64 

14 

17 

77 

0 

0 

74 

19 

Jl 

1  + 

3 

11 

66 

13 

9 

64 

2 

9 

74 

11 

11 

7  + 

6 

14, 

14 

10 

3 

71 

2 

13 

]9 

11 

14 

105 

9 

12 

13 

17 

5 

AVOIRDUPOIS  WEIGHT. 

16  Drains  (dr.)  make  1  Ounce,  oz. 
16  Ounces     — -   1    Pound,   lb. 


oQ    Pftiinfia                 .  -    ^    of  a  htinc 

1.  qr. 
.  Cwt. 

4  Quarters   T  Hundrt'd 

20  Hundred 1    Ton,  T. 

Note. — By  this  weight  are  vveighevi  all  kinds  of 

coarse 

and  heavy  Goods,   except  Gold,   Silver, 

&c. 

!b.     oz.    dr.       tons.  cwt.  qr.  lb. 

lb. 

oz. 

dr. 

1.318"     10     10     2.416     19     2     26     3, 

.539 

13 

15 

436       9        8          313      10     0     20 

316 

14 

13 

62A     14       6         2   1      11     3     1& 

223 

\2 

7 

419       6      15          725      19     ^      18 

811 

9 

6 

245        9        7          357      14     2     25 

700 

6 

14 

853     11      10         429      17     3     22 

414 

12 

12 

145       9        8         235     15     2     19 

0 

0 

0 

toJis.   cwt.  qr.  lb.         tons.  cut.  qr. 

cwt. 

qr. 

lb. 

4.    305     14     2     11     5.    174     19     3     ( 

5.  174 

3 

27 

418      18     0       0              74      14     2 

724 

2 

24 

336       2     1      14           714     13     1 

149 

1 

14 

119      13     3     27            718      16     2 

719 

2 

16 

767     16     0       8           734     15     2 

407 

1 

23 

782       9      1      16            714     14     1 

149 

2 

17 

421      15     3     19            155       0     3 

7ti 

3 

15 

^2  COMPOUND    DIVISION. 


qr. 

lb. 

oz. 

lb. 

oz.  drs. 

7-  44 

27 

15 

8.   17 

15     15 

74 

26 

14 

27 

14     11 

19 

14 

13 

16 

13       9 

74 

12 

14 

74 

14      14 

66 

27 

13 

70 

0        0 

74 

19 

10 

04 

13      10 

13 

17 

5 

13 

4       5 

APOTHECARIES'   WEIGHT. 

20  Grains  (gr.)  make   1  Scruple,  9. 

3  Scruples 1  Dram,  5* 

8  Drams         —  1  Ounce,  ^. 

12  Ounces 1  Pound,  j^ 

lb.  oz.  dr.        oz.  dr.  so.  gr.  lb.  oz.  dr.  sc.  gr. 


1.314     8 

4 

2.  22     3     2 

19 

3.  646 

U 

4 

1 

19 

210  11 

4 

56     0      1 

13 

715 

3 

7 

1 

14 

766    10 

2 

43     2     2 

11 

934 

3 

4 

0 

12 

555     9 

6 

54     7      0 

17 

373 

10 

5 

2 

9 

417     8 

1 

76     5     2 

14 

216 

5 

1 

2 

16 

324     7 

3 

45     6      1 

0 

159 

2 

5 

0 

14 

lb.   oz.  dr. 

oz.  dr.  sc. 

dr. 

sc.  gr. 

lb. 

oz.  1 

dr. 

4.  47    11      7 

5 

. 149  7      2     6 

.749 

2  19 

7. 

84 

11 

7 

94    10     6 

714  3     0 

607 

1    18 

74 

10 

6 

74   10     4 

619   2      1 

714 

2   17 

37 

5 

4 

75     9     3 

74  6     2 

400 

0     0 

19 

4 

3 

69     0     2 

162  5     2 

74 

1    13 

74 

1 

2 

57      1      2 

74    1     2 

715 

2    14 

79 

2 

6 

18     2     1 

779  6      1 

64 

1    18 

19 

2 

4 

19     3     5 

146   4     0 

16 

0   10 

13 

4 

8 



, 

COMPOUXD    ADDITION'.  53 

CLOTH  MEASURE. 

2J.  Inches  (In.)  make    1  Nail,  ni. 
4    Nails.  I  of  a  yard,  qr. 

4  Quarters         1  Yard,  yd. 

3    Qufirters         1  Ell  Flemish,  E.  Fl. 

5  Quarters         I  Ell  English,  E.  E. 

6  Quarters 1  Ell  French,  E.  Fr. 


yd. 

qr.  nl. 

K.e.  qr.  nl. 

E.e.  qr.  nl. 

yd.qr.  nl, 

1.  4:i4 

3  2     2. 

,511   4     2 

3.565  4     0     4. 

.543  3  2 

527 

1   ? 

660  2     0 

626  2      I 

836  2  2 

613 

2  3 

439   4     2 

724  0     1 

754  2  3 

758 

3   1 

337    I      2 

882  2      3 

217    1    3 

840 

1    3 

854  2     3 

933  0     3 

725  3   2 

925 

2  2 

766  0     2 

227    1       1 

438  2  2 

E.e. 

qr.  nl. 

E.e.  qr.  nl. 

yd.  qr.  nl. 

E  e.  qr.  nl 

5,  120 

2  2     6. 

. 537    0     2 

7.   74  3     S     8. 

.    77'  4  3 

3P4 

4   1 

916  3      1 

64  2     1 

14  3  2 

110 

2  0 

328  3     3 

74    1      3 

74  2   I 

481 

1    2 

457    I      2 

49  2      I 

49    1    2 

556 

4  3 

646  3     2 

74*  1      2 

74  2   I 

664 

3   I 

2^7   4     2 

44  3      / 

44    I   2 

779 

2  3 

561    2     2 

16  2     3 

94  0  2 

LONG    MEASURE. 

3 

Barley  Corns(bc.)n] 

lake   1     Inch,  in. 

12 

Inrhes 

- 

1      Foot,  ft. 

1G» 

Feet 

-i 

1     Rod,r. 

40 

Rods 

- 

1     Furlong. 

,  fur. 

8 

Furlonc;s 

1      Mile,  m. 

69^ 

Statute  Miles 

— -  1     Degree, 

Deg. 

5 

i* 

^4  eOMPOUND     ADDITIOI?. 

ALSO, 


4     Inches 

make  1     Hand, 

3     Feet 

I     Yard, 

5f    Yards 

1     Hod,  Pole,  or  Perch. 

6     Feet 

1     Fathom. 

66     Feet 

— .—  1     Gunter's  Chain. 

S     Miles 

EXAMPLES. 

miles,  fur.   p. 

yds. 

yds.  ft.  in.  be.      lea.  mi.  fur.  p. 

1.  427     6    23 

3     2.  214  2     9    0     3.  320    1      6      IS 

689     5    26 

5 

183  2   11     2          623    1     7     27 

322     7     30 

2 

597  0     8     1           721    0     4      I6 

510     2    38 

4 

649  2     7    2          826    1      3     32 

777     4      0 

3 

725    16     1           932   2     G       1 

888     3     10 

4 

930   13    0           315    1      2     28 

]26     0    ^^4 

0 

492   1      4     1           409    1      5     39 

412     7    39 

4 

291   2   10    2          376   2     7     27 

lea.  m.  fur. 

fur 

.  p.  yds       p.  yds.  ft.       feet  in.  b.e 

4.   17  2     7     5. 

147 

39  5     6.  177     5  2     7.  174  11     2 

14   1     6 

614  37   4          714     4    I              49    10     1 

74   1     7 

714 

193         714     12             74   11     2 

68  2     4 

674 

17    1          615     0   I             64     9      I 

74  1     0 

719 

27  2         714     1   2            74  10     I 

69  2     1 

197 

19  I          719     1    1             64   11      2 

74  1     2 

724 

14  3         437     2  1             74   10     0 

96  2     4 

604 

29  5          610     4  0             94  n      2 

LAND,  OR  SQUARE  MEASURE. 

144  Square  Inches  make  1   Square  Foot. 

40      Rods 1      Rood. 

4      Roods    I     — —  Acrtfi 


COMPOUND    ADDITION.  ^5' 

ALSO, 

9  square  Feet  make  1  square  Yard. 

sol    Yarda  — ■  I Rod. 

160      Rods  I  ^cre. 

640      Acres  1  Mile. 


EXAMPLES. 

ac. 

r. 

P- 

ac. 

r. 

P- 

ac.   r. 

P- 

1.  452 

2 

38 

2.  982 

2 

24 

3. 

921  1 

29 

114 

1 

35 

618 

3 

14 

604  3 

32 

715 

2 

16 

100 

1 

27 

736  2 

29 

430 

2 

35 

474 

2 

10 

559  3 

28 

529 

3 

7 

363 

1 

31 

265   1 

17 

2 '1-6 

1 

23 

755 

3 

38 

427  0 

30 

6fil 

3 

11 

647 

0 

6 

883   I 

39 

314 

2 

35 

234 

2 

29 

P- 

291   3 

25 

ac.  r. 

P 

ac.  r.  p. 

ac.  r. 

ac.  r 

•  P- 

t.  17   3  39 

5. 

714  3  39 

6. 

14  3 

39 

7.  174  3 

39 

64  2 

37 

619  1  36 

74  1 

19 

714  1 

27 

74  1 

U 

714  2  27 

64  2 

14 

618  2 

12 

64  2 

19 

619  1  34 

74  1 

18 

719  1 

14 

74  1 

18 

719  2  37 

4*7  2 

24 

734  2 

II 

64  2 

17 

719  1  24 

18  1 

14 

715  1 

24 

14  1 

13 

615  2  14 

74  2 

19 

639  2 

24 

94  3 

54 

174  3  38 

74  2 

24 

714  1 

34 

'_ 

LIQUID  MEASURE. 

4  Gills    (gl.),        make  1  Pint,  pt. 

2  Pints  I  Quart,  qt 

4  Quarts  I  Gallon,  gal. 

63  Gallons  1  Hogshead,  hhd. 

2  Hogsheads  I  Pipe,  P.  or  Butt.  B. 

2  Pipes  or  Butts    — ^  1  Tun,  T, 


56  e-^MPOUND    ADDITION. 


hhd.^al.  pt./ 

tuns,  h.  g.  qt 

tuns,  h.  g.  q. 

1.  626  U  7 

2.  522  I  39  3 

3.  148  2  25  3 

753  17  1 

257'  S  34  2 

513  0  42  3 

4:8  5^  6 

763  2  58  3 

614  1  S6  1 

217  1.^  7 

611  3  43  1 

349  3  43  2 

1.33  1^5  0 

937  1  16  3 

416  2  56  1 

A97   56  2 

23S  0  31  2 

952  3  26  0 

J^312  11  3 

749  3  7  0 

567  1  19  3 

W   256  <>  0 

SI9  2  59  3 

792  3  46  2 

• 

tun^.hlid.s:. 

hh«l  ^al  qt. 

p:.  q-  p- 

4.  7 1 4  3  62 

5.  74  41  3 

6.  14  3  1 

614-  2  61 

64  40  3 

74   I 

174  1  39 

74  19  1 

39  3  1 

164  2  47 

64  39  2 

17  1  0 

274  1  49 

74  40  I 

19  2  0 

175  -2  37 

69  16  I 

77  1  1 

375  I  i9 

17  39  2 

39  3  1 

704  n  64 

28  44  3 

24  ^2  0 

V 

DRY   MEASURE. 

2  Pints  (pt.)  make  I  Quart,  qt. 

4  Quarts         1  Gallon,  g;al. 

2  Gftllons       >-  '  I  Peck,   pk. 

4  Pecks  1  Bushel,   bu. 

40  Bushels        1  Load,  Lo. 

bu.  pk.  gal.      bu.  pk.  8:al.      bu.  pk.  gjal. 
1.73     3     1     2.29     2     0     3.36     2     1 


46 

2 

0 

57 

0 

1 

99 

3 

I 

39 

3 

1 

38 

3 

1 

36 

3 

1 

48 

2 

0 

26 

2 

0 

•27 

2 

1 

37 

2 

0 

48 

1 

0 

46 

3 

0 

46 

1 

I 

28 

0 

1 

27 

2 

1 

27 

Q 

1 

76 

3 

I 

36 

1 

1 

39 

I 

1 

24 

2 

1 

57 

2 

1 

COMPOUND    ADDITION,  57 


gal.  qts.  [)ts. 

pks.gal.  qts. 

bu. 

pk.  gal.  qts. 

4.  56  3  1 

5.  76  1  3 

6.  58 

3  1  3 

77  2  1 

39  0  2 

74 

2  1   3 

64  1  0 

92  1   3 

63 

3  0  2 

76   1   1 

47  0  3 

49 

1   1   3 

67  2  1 

36   1   2 

48 

2  1  3 

74.  3  0 

27   1   3 

63 

3  1   3 

62  1   1 

64  0  0 

15 

0  0  0 

^9  2   1 

77   I  2 

3o 

3  0  2 



TIME. 

60     Seconds  (Sec.)  make  1  Minute,  ra. 

60     Minutes  1   Hour,  li. 

24     rtours  1   Day,  d 

3651    l)ays  1  Year.  yr. 

100     Years  1   Century,  Cen. 


mo.  w. 

d. 

h*    w. 

d. 

h. 

"mi. 

d.  h.  mi. 

sec. 

1.  19  2 

6 

19  2.57 

4 

23 

38  2 

.62  7  47 

38 

46   1 

4 

21    64 

6 

13 

47 

18  12  54 

56 

22  3 

5 
S 

9    15 

3 

21 

18 

19 
15 

76  21  16 

34  9  ^ZO 

49 

57     2 

21    36 

2 

31 

62   1 

ii 

12    78 

6 

9 

59 

bO  23  3i 

45 

17  3 

2 

14    49 

0 

20 

*  6 

52  22  28 

32 

11   3 

4 

16    71 

5 

14 

48 

15  4  5S 

23 

29   I 

3 

21    23 

3 

7 

24 

64  I6  13 

16 

yrs.  mo. 

w. 

uio.  w.  d . 

d 

avs, 

h.  m. 

hrs.  min. 

se. 

4.737  12 

3 

5.  64  3  6 

6. 

714 

23  59 

7.  647  59 

59 

347  11 

2 

74  1  5 

74 

14  54 

137  51. 

54 

618  10 

1 

34  2  8 

94 

21  55 

375  56 

56 

374  9 

2 

74  1  4 

74 

13  53 

714  17 

19 

175  I 

1 

63  2  1 

69 

12  14 

615  54 

54 

714  12 

3 

74  1  2 

74 

12  19 

714  17 

13 

615  10 

I 

64  2  1 

37 

11  17 

613  34 

56 

314  9 

3 

94  2  6 

46 

22  49 

626  47 

49 

58  COMPOUND     ADDITION. 

A8  rR0N(3xMY. 

60  Seconds  (")  make  1   Prime  Minute,'. 

60  Minutes         1   Degree,  °. 

30  Dei^rees 1  Siun,  S. 

12  Sij2;nsor>      __  C  The  jjjreat  circle 

360  Degrees^  I    oftheZodiack 


11 

24 

37 

41 

5   3 

26 

25 

6 

9 

54 

35 

7 

12 

5- 

21 

9   5 

37 

56 

3 

29 

59 

7 

3 

25 

13 

17 

8  24 

42 

59 

11 

26 

21 

19 

4 

29 

18 

29 

3  9 

12 

15 

9 

24 

50 

40 

5 

16 

52 

43 

4  8 

17 

41 

11 

18 

29 

27 

3 

19 

47 

51 

S2(^ 

9 

8 

5 

13 

51 

46 

11 

29 

51 

36 

5  16 

8 

27 

6 

7 

1 

9 

9 

18 

30 

SO 

11  20 

40 

50 

10 

12 

24 

35 

3 

4 

44 

44 

10  9 

55 

•  >7 

7 

21 

42 

.'=6 

7 

25 

36 

51 

4  21 

4-4 

56 

5 

^23 

51 

46 

MISCELLANEOUS  EXAMPLES  IN  ADDITION. 

1.  What  is  the  sum  total,  in  shillinu;s,  ot  54  guineas, 
29  pounds,  36  guineas,   and  48  pounds? 

Answer,  3430  shillings. 

2.  Add  together  16/.  12s.  2(1. ;  156/.  9s.  9l(l.;  20395/ 
12s.:  24/.  19s.  lljd. ;  37/.  6s.  Id.;  327/.  18s.;  and 
100  iTuineas.  Ans. 21063/.  18s.  6d. 

3.  In  collecting  an  account  of  debts  owing  to  «ne,  I 
find  Mr.  A.  owes  me  74D.  16cts. ;  Mr.  B.  69 D.  50cts. ; 
Mr.  C.  73!>.  4cts.  ;  Mr.  D  38!3.  37^cts. ;  Vlr.  E.  14D. 
e^cts.  ;  what  is  the  whole  sum  due  to  me  } 

Ans.  26?  ).  13lcts. 

4.  A  gentleman  ordered  a  service  of  plate  from  his 
silversmith,  anil  on  receiving  his  bill,  be  fin  Is  that  he 
had  dishes  and  covers  weighing  45  lb.  9  oz.  12  dwts. ; 
plates  weighing  70  lb  7  oz!  16  dwts.  ;  spoons  of  difter- 
ent  sizes,  and  ladles,  24  !b.  9  ;>z.  12  Iwts  :  waiters, 
15  lb.  10  oz.  ;  salts  and  castors,  4  lb.  4  oz.  3  dwts.; 
candlesticks,   19  lb.  11  oz.  17  dwts. ;  and  sundry  snial- 


COMPOUND    ADDITION.  59 

ler  articles  5  lb.  3  oz.  10  clwts. ;   what  is   the  weight  of 
silvtT  he  will  hiv  •  to  pay  for  ?      Ans.  l86ib.  Soz.  iOitwt: 

5.  A  carrier  hritie;s  goods  to  a  shop  k«^eper,  viz.  8  h;i<;s 
of  hops  wei^hioi;  19  cwt  3  qrs.  14  lb.  :  cheeses  wei}>;hing 
J 5  rwt.  1  qr.  21  lb.  ;  butter  wei^jhino;  12  cwt.  2  qra.  ; 
two  chests  of  tea  weighing;  ]l  cwt.  each  ;  and  a  sack  of 
salt  weighing  8  cwt.  2  qr.  12  lb.  ;  how  n>uch  weight  wdl 
the  earner  have  to  cliarge  t'     Ans.  59  cwt.  1  qr.  19  lb. 

6  The  lent  of  my  house  is  30/.  per  annum  :  the  house 
taxis  three  pound  fifteen  shillings  ;  land  tax  51. ;  vvin- 
dous  15/.  I2.S'.  Oc?.  ;  poor's  rates  10/.;  lightincr,  watch- 
ing, and  street  rates  3/.  9s.  3ld. :  how  much  therefore 
do  my  house  and  taxes  stand  n»e  in  per  annum  ? 

Ans.  87/.  lijs.  3ld, 

7.  'I'he  following  is  an  estimate  of  the  repairs  wanting 
to  my  house;  how  much  is  the  whole  sum  ?  ('arpenter*s 
bill  27/.  9s.  9(11,  ;  bricklayer's  and  plasterer's  17/.  7s. 
ad.  ;  mason's  5/.  5s.  ;  painter's,  glazier's,  and  plum- 
ber's, fourteen  gu.neas  ;  smith's,  for  new  rails,  12/.; 
and  the  slater's  9/.  18.s.  Ans.  86/.  14.>-.  3^. 

8.  A  man  purchased  some  goods  for  the  country  ;  the 
first  parcel  contained  25  y«ls.  2  qr-  2  nl  of  broad  cloth  ; 
the  second  \26  yds.  2  qrs.  of  serge  ;  the  tisird  a  thousand 
yards  of  green  baisc  ;  and  the  fourth  19  yds.  3  qrs.  2  nl. 
of  shadoon  ;   what  was  the  whole  quantity  ? 

Ans.   1172  yds.  0  qr.  0  nl. 

9.  A  wine  merchant,  retiring  from  business,  takes  an 
account  ot  the  stock  of  wines  in  his  cellar,  and  finds  6 
pipes  and  50  gallons  of  port  wine,  four  pipes  of  sherry  ; 
ten  pipes  of  Lisbon  ;  2  pipes  of  claret  ;  of  Madeira  he 
hail  36  gallons  ;  of  brandy  50  gallons  ;  of  rum  two  hogs- 
heads ;  and  of  Holland,  I  hhd.  and  12  gallons;  what 
quantity  of  liquor  dia  his  cellar  contain  ? 

A  us.  23  pipefe,  1  hhd.  22.  gal. 

10.  \  friend  in  Essex  desired  me  to  measure  his  farm, 
which  he  holds  on  a  lease  ;  t[»e  three  fields  at  the  back  of 
the  house  measured  59  ac.  2  r.  20  p.  ;  the  large  piece  of 
ground  in  the  valley  measures  74  acres,  three  others 
measure  each  on  an  average  I  \  ac.  1  r.  36  p.;  the  field 
laid  down  in  clover  contai^is  7  ac.  3  r.  2  p.  one  sosvn 
with  caraways,  1  find  to  be  3*  acres ;  and  the  ground  be- 


60  COMPOUND    SUBTRACTION. 

longing  to   the  garden,    out-houses,  &c.   makes   about 
li  acres  ;  how  many  acres  ought  he  to  pay  for  ? 

Ans.  180  ac.  2  rd.  10  per, 

1 1.  A  merchant  sends  to  his  banker  on  the  2d  day  of 
the  month,  in  money  and  bills,  to  the  amount  of  two 
thousand  guineas  ;  on  the  fifth  he  sends  him  900/.  I9s, 
4-d, ;  on  the  eleventh  he  sends  500/. ;  and  in  the  course  of 
the  remaining  days  of  the  month  he  sends  1515/.  I2s. 
Hid.;  how  much  therefore  may  he  draw  as  occasion 
requires  ?  Ans.  6016/.  I2.s.  3ld. 

12.  A  gentleman's  steward  received  the  following 
sums  of  money  for  rents  ;  what  was  the  gentleman's  in- 
come ?  Of  farmer  A  he  received  394/.  12s.  6d.,  of  B  971. 
Us.  9rf.,  of  C  175/.  10s.,  of  D  99/.  4s.  and  of  K  139/. 
12s.  46/.  Ans.  906/.  I3s.7rf. 

13.  A  person  borrows  of  several  friends  the  following 
suujsof  money  :  of  the  first  500/. ;  of  the  second  225/. 
12s.  ;  of  the  third  fifty  guineas  ;  of  the  fourth  seventy 
guineas  and  22  crowns  ;  of  the  fifth  he  had  150/.  7s.  6d.i 
how  much  will  he  have  to  pay  interest  for  ? 

Ans.  1007/.  9s.  6d, 

14.  A  man  borrowed  a  sum  of  money,  and  paid  at  dif- 
ferent times  87  dollars,  but  he  still  owed  64  D.  372  cts., 
what  was  the  original  debt  ?  '  Ans.  15 ID.  37aCts. 


COMPOUND  SUBTRACTION, 

Is  the  method  of  finding  the  difference  between  two 
given  compound  numbers. 

Rule.  I.  Having  arranged  the  numbers  so  that  the 
smaller  may  stand  under  the  greater,  subtract  each  num- 
ber in  the  lower  line  fiom  that  which  stands  above  il;, 
and  write  down  the  remainders. 

2.  \\  ben  any  of  the  lower  denominations  are  greater 
than  the  .;pper.  increase  the  upper  numuer  by  as  ujany 
as  make  one  of  the  next  superior  denomination,  from 
which   take  the   figure  in  the   lower  line,   set    down 


bOMPOUNP     SUBTRACTION. 


61 


the  difference,  and  carry  one  to  the  next  number  in  the 
lower  line,  and  subtract  as  before. 

Ex.  Subtract  595/.  ITs^^ld.  from  600/.  lOs,  1\d. 

Here  I  say  2  farthings  from  1, 
I  cannot,  but  I  add  4  to  the  1,  be- 
cause 4  farthings  inak*^  a  penny, 
and  2  from  5,  and  there  reinains 
\  ;  1  carry  one  to  the  9  ;  10  fro^ 
7  I  cannot,  but  I  add  12  to  7,  be- 
Proof  600  10  1\  cause  J2  pence  make  a  shilling, 
and  10  from  19  and  there  remain 
9  ;  I  carry  1  to  17  ;  and  18  from 
10,  T  cannot,  but  1  add  20  to  the  10,  because  20  shillings 
make  a  pound,  and  18  from  30  and  there  remain  12;  I 
now  carry  one  to  the  five,  and  go  on  as  in  simple  sub- 
traction. 

The  method  of  proof  is  the  same   as  in   simple  Sub- 
traction. 


600 

595 

10 

17 

d. 

9^ 

4 

12 

^\ 

600 

10 

7i 

EXAMPLES. 

D.    cts.  rals.      D.    cts.  mis.  D.  cts.  mis. 

Ex.  1.39     44     3       2.76     29     4  3.18  7e»     5 

27     7G     2           49      13     6    *  9  47     6 


yJU    rsrwT 


D.    cts.  mis.       D.     cts.  mis,       D.   cts.  mis. 
Ex.  4.  57     13     7        5.62      13     7        6.48     30     1 
37        4      1         •    24     97     % 


49     76     9 


fwrr    TjtTT 


Ea.  D.  D.  cts.  mis. 
Ex.  r.  67     S     7     4     6 

39     4     2     9     7 


/7ir"F 


TSJL 


Ea.  D.   D.  cts.  mis. 
8. 79     4     I     6    7 
37     6     7     4     9 

{rrcTri: 


6£ 


COMPOUND    SUBTRACTION. 


L.    s.      d.  L.     s.  d.  L.     s.  d» 

Ex.  9.  145  19     9*  Ex.  10.370  17  H  Ex.  11.450  12  61 
130   17     6^  369   12  4^  371    10  4 


Answer 
Proof 


L.     s,    d.  L.     s.  d.  L.     s.   d. 

Bit.  12.  594   10  91  Ex.  13.465   12  7^Ex.  14.564  12-2* 
374   19  51  349   17  91  37d   18  4^ 


L.      s,    d. 

Ex.  15.371      19     2| 

199     17    1]^ 

L.    s. 
Ex.  16.  700    0 

376    16 

d, 
0 

L.     s.     d, 

Ex.  ir.  47d     19     4 

374     12     9 

L.     s. 

Ex.  18.473   18 

291    12 

d. 

L.     s.    d, 

Ex.  19.24j^     9     9^ 
159   19  Hi 

L.      s. 

Ex.  20.  376  17 
299    14 

d. 

7 
4| 

L.     s.     d.      ' 

Ex.  21. 594     0     0 

593    19     9^ 

L.     s. 

Ex.  S2.  796  12  ] 

^69     8 

d. 

'^    r-  •  j'\'               ../ 

%f^f 

L.    s.     d, 

Ex.  23.476    17     7 
•      3^9    19    111 

Ex.  24.  399     2 
177  12 

d. 

2 

n 

\  \  r  ' « \i 


COMPOUND    SUBTRACTION. 


63 


L. 

Ex.  25.  209 

159 

s.     d. 

18  8 

19  9^ 

Ex. 

Ex. 

Ex. 

Ex. 

Ex. 

Ex 

Ex.: 

Ex.4 

L. 

26.  500 
499 

0     0 
19  11 

L. 

Ex.  27.  422 
371 

s.     d. 
3     6* 

15     7^ 

28.  224 
156 

s.     d. 
2     6| 
6     6^ 

L. 

Ex.  29.  794 
367 

15  ()l 

16  4^ 

L. 

,  30.  999 
[800 

0     0 
19   111 

_ 

L. 

Ex.  31,  704 

398 

.s.       d, 

15     4^ 
12   11 

L. 

S2.  074 
249 

s.     d, 

6     02 

19     91 

■"  "    " 

L, 

Ex.  33.  372 
U9 

s,     d. 

10     61 

6     4i 

L. 

34.  649 

597 

s.     d. 
12     91 
19     81 

L. 

Ex.35.  341 

230 

s.     d. 
5   llj 
9     4^ 

,  36.  846 
375 

s.     d. 
9     81 
9     9^ 

L. 

Ex.  37.  124 

109 

^.     J.. 

9  10^ 
10     3* 

L. 

J8.  90441 
67217 

5     91 
13   10 

L. 

0.  12i27 
7618 

L, 

Ex.  39.  418 
399 

s.    rf. 

7    10 

16     9^ 

s.     d. 
16   lU 
14     91 

64 


COMPOUND  SUBTRACTION. 

L,     s.    a.  L.  s.  d. 

Ex.  41.1654  12     7  Ex.  42.  14476  5  6* 

585     9  10^  7iJ4  13  8| 


L.     s.     d.  L.       s.     d. 

Ex.  43.222    18     9^         Ex.  44.  96481    16     9 
i42     7  10^  37(>8    10     91 


X.    s.     d,  L.      s.     d. 

Ex.45.  164  17     8^         Ex.46.  18149    14     0* 

29     2     91  ]7'2\6     0     4j 


L.     s,     d.  L     s.      d. 

Ex.  47.  417     4  10^         Ex.  48.  20412  13     9| 
319  11     7\  19911-   14     tl 


L,    s.    d,  L,        s.     d. 

Ex.  49.425  18     9  Ex.  50.  22425   14     9* 

139    10     9|  21018*   8  Hi 


L.    s.    d.  L.      s.    d- 

EX.5U183     9     IJ  Ex;52.  24463   13  HI 

24   14  10*  177S2  iQ     o» 


L.     s.    d.  L.      s.     d. 

Ex.  63.  42116  91         Ex.*  54.  86476  6  9^ 

326  19  0  56117  13  10 


L.     s.     d.  L.       s.    d, 

Ex.  55.  433  17  2|    Ex.  56.  28446  17  9 
311  19  41  19994  14  8| 


COMPOUND  SUBTRACTION.                               6^ 

L.     s.     d,  L.  s.  d. 

ES.  57.  194   12     8i  Ex.  58.  80490  9  9 

117    12     9  2^089  15  \0l 


L     s.     d.  .     L.       .s.     d, 

Ex.  59.  474    19     4^  Ex.  60.  26475    13     9 

3t>2    ]3     ll  247  10    18    Ml 


L;      s.     d.  L.       s.     rf. 

Ex.61.  4559   16     91         Ex.  62.  3U87    15    11^ 
3228     9     5i  31767    19    lO 


I.         s.     d.  L.         s.     tZ. 

Ex.  63.     2139       7     10  Ex.     64.     56492     7     5^ 

1914      13      10* 


L,         si     d.  L.  a      d, 

Ex.  65.     3471      19      9i  Ex.     66.     38410  14     9 

293     19      9^  28019   19    10^ 


if. 

Ex. 
Ex. 

Ex;  67. 

L. 

4557 

3945 

s. 
18 
17 

d. 

68.  601273 
462104 

>.  d, 
11  7 

1 5  8] 

• 

E3^  69. 

L. 

5:34. 

559 

.s. 
11 
12 

d, 

3 

7 

L. 

70.  424156 

379 1 2*  i 

s.      d. 
11   6i 
10  9^ 

• 

L.       s.     rf.  L.         s.     d. 

Ex.   ri.     7  860     0     0  Ex.  72.     44I.J91      (3     0* 

32.  I      4.     7  '  38909  i      9     8^ 


66  COMPOUND    SUBTRACTION. 


L.       s.     d. 

Ex.  72>.  6234     6    6 
309   12   lOi 

Ex.  74.  1414     9     91 
7c9    12  111 

L.      'S.    d,  L.      s.     d, 

*Ex.r5.  1173   14     9|       Ex.  76.484760   10     9 
437    18   11*  329189    19     9l 


Ex. 

Ex. 
Ex 
Ex.  84 

Ex. 
Ex.  88 

'    ^ 

L.     s.     d. 

Ex.  77.  791     5   11^ 
261    19    11* 

L.       s.    d. 

78.  14112     0     0| 
4612  19      1 

L.       s.     d. 

Ex.  79.  1345    19     9^ 
34.5  17     91 

L.     s.      d, 

80.  4621    15     9l 
S94   1 J     0| 

L.     s.     d.     . 
fix.  81.396  19     9\ 
29    19     9^ 

•       L.     s.     d. 

:.  82.  254  14     9^ 
244   19    10| 

.   ^ 

L.     s.     d. 

Ex.  83. 1214     0     5 
880     0     8^ 

L         s.    d. 

.  564121    10  I0| 
379178    16   lOi 

L.      s.      d, 

Ex.  85.  4465  10     9* 
304     0  \\\ 

L        s.    d. 

86  453  2   13     9l 
4319    15    n| 

L.     s.     d. 

Ex.  87.408   19     4* 
254     I    10^ 

L.      s.    d. 

.60985-14     4-1 
1427    19     91 

COMPOUND    SUBTRACTION. 


6t 


L.    s.    d. 

L.    s.    d. 

Borrowed  SOO    o    0 

Borrowed  ] 
i 

1000     0     0 

(    \9  1.5      0 

'577    16     7* 

Paid  at    \   S9     7     71 

Paid  at    ^ 

105     0     3 

different  J    76     8     1 

different  < 

'    52   10    11 

times      j    4d   15   10 

times     i 

2i6     9     9* 

(^105     0     0     . 

( 

Paid    -~ 

..'JOG     0     9 

Paid      -     283     6     6| 

881    18      4 

Unpaid.     16   13     51 

Unpaid     - 

M8      1      8 

Suppose  a  person  is  debtor 

And  is  creditor,  by  book- 

to  sundry  persons  in  the 

debts  from  different  peo- 

following sums. 

ple,  in  the  foilowiny;  sums. 

L.     s.      d. 

L. 

s.      cf. 

r678    14     9^ 

764 

14     9| 

2*   17      4i 

39 

14     4 

5     5     0 

500 

0     0 

1054   12     91 

8V9 

5     95 

26     5     0 

2500 

0     0 

7     7     0 

5503 

5  n 

95   19     9* 

30(Kt 

^J     0         , 

3'J   11     S 

Br. 

CK 
Dr. 

Balance  in  favo 

ur  of  Cr. 

Required  the  balance  of 

Required  the  ha'anceof 

this  account  ? 

tills  account  ? 

Dr.               .  Cr. 

Dr. 

Cr. 

L.     s.    d,        L.    s.    d. 

Z.    s.    d. 

L.    s.  rf. 

764   <4     9       39r    14   111 

769    19    \0l 

49  12  11 

397     0   10^      2()7   Ij     9 

643     4     4 

1000   17     91 

.2M>  19     9^      72i)   13     8^ 

24s   11      7 

iro6     5     5 

467    16     7^      464   Id     0 

591      8     4 

4     4     O 

371    14     9       215    12     6. 

<     19     6 

250    12     8| 

564   12     6|      345     Q    \L)\ 

3^0     i;     0 

175?     17     0 

&S  COMPOUND    SUBTRACTIOIS'. 

EXAMPLES  OF  WEIGHTS  AND  MEASURES, 

TROY   WEIGHT. 

lb.  oz.dwt.  gr.  Ih.  oz.dwt.sr.       lb.  oz.dwt.gr 

Ex.  1.  187   9    12     20     2.256  6  0     22  3.567     4     0  0 

169   6   14      17  199  9  3     20         379    i  1      9  9 


lb.    oz.  (Iwt.i!;r.        lb.    oz  dwt.  gr.        lb.    oz.  dwt.gr 

4.254     0     0     0     5.673     3     0     9     6  423     5    15      14 

25.1    11    19   20  567     9    17    16  246    '1     18     23 


lb.  oz  dwt.  oz.dwt.gr.    lb.  oz.  dwt.  oz.dwt.gr. 

7.14  II   9  8.  74  12  18  9.175  3  10  10.17  10  20 

11  10  14    71  14  17    159  11  14     14  n  33 


AVOIRDUPOIS  WEIGHT. 

tons;  cwt.  qr.  lb.  oz.  dr.   tons.  cwt.  qr.  lb.  oz.  dr.^ 

1.72  10  3   14  10  12  2.  64  15  2  15  JO  9 

•  9   16  1  25   14   6    46   15  3  5  12  14 


tons.  cwt.  qr.  II).  oz.  dr.     tons,  cwt.  qr.  lb.  (\z   dr. 
3.  25  0  0  0  0  0     4.  67  2   1  4  14  2 
24  0  2  0  0  15       29  14  3  -2   0  14 


tons,  cwt.  qr.  lb.  t)Z.  dr;     tons.  cwt.  qr.  II;.  oz.  dr. 
5.  36  7   1   1   1   1      6.  76  3  0  0  0  4 
30  3  2  5  5  5         67  12  2  0  14  4 


tons,  cut.  qr.  cwt.qr.  lb.    qr.  lb.  oz.    lb.  oz.  dr. 

7.  14   12  2  8.  17  I  25  9.  143  22  12  10.  174  II  10 

1   14  3    14  2  27   .  74  19  14     39  12  13 


COMPOUND    SUBTRACTION.  Q9L 

APOTHECARIES  WEIGHT. 

Ih.  oz.  dr.  scr.  lb.,  oi.  dr.  scr.      lb.   oz.  dr.  scr. 

I.  4-56     9     4     0      2.  269     8     3     2      3.  987     4-4     0 

399     4     7.    2  178   11      3      1  379      10     5     I 


lb.  oz.  dr.  scr.        lb.   oz.  dr.  scr.         lb.    oz.dr.scr. 

4.564     0     0     0      5,  375     7     7     I       6.  394     2     2     0 

469     3     3     2  369     4     7     2  299  11     7     2 


lb.     oz.  dr.        oz.  dr.  scr.       dr.  scr.  gr.        lb.  oz.dr 

7.  144      10     6     8.  27     4     1     9.  27     1    14   10.  74  10  5 

64     11     7  14     7      2  14     0  19         65   1 1   6 


CLOTH  MEASURE. 

yds.  qr.  n.     E.e.  qr.  n.       yds.  dr.  n.  yds.  qr.n 

Bx.  1.218  2     0  2.46     0     0  3.567     I      1  4.459  12 

167  1  3   23  2  2   469  0  2  399  3  3 


yds.  qr.  n.    E.e.  qr.n.   E.e.  ^qr,  n.   E.e.  qr.  n. 
5.  174  2  1  6.  174  3  1  '  7.  171  1  3  8.  J2  1   I 

39  3  2     49  4  2     74  4  2    10  4  3 


LONG  MEASURE. 

yds.    ft.  in.  b.c.       yds.  ft   in.  b.c.  yds.ft.in.be. 

Ex.  1.456     2     10     1      2.669  0      0     0  3.267    1    1     1 

379  1  11  2    599  1   I   1  199  2  2  2 


lea.  n^.  fur.  p.    lea.  m.  fur.  p.    lea.  m.  fur.  p. 

4.470  1  4  19  4.  367  0  0  0  6.  225  1   11 

279  2  7  23    179  2  5  23     167  2  4  4 


*  70  COMPOUND    SUBTRACTIOK. 

lea.  m.  fur.     fur.    p.  yds.      p.  yd.  ft.  ft.  in.  b.c. 

7.21     2     4     8.14     34     5.    9.14     3      1  10.17  11     2 

3     2     6         12     39     5           9     4     2  14   11      1 


LANF)  MEASURE. 

ac.  r.  p.    ac.  r.  p.    ac.  r.  p.  ac.  r.  p. 

Ex.  1.456  2  25  2.4.57  1  29  3.356  0  39  4.594  I  1 

Syy  0  29   2b4  3  39   279  3  39  259  3  17 


ac.  r.  p.    ac.  r.  p.    ac.  r.  p.    ac.  r.  p. 

5.  12  0  32     6.  112  I  31  7.  12  I  25  8.  19  1  20 

1  3  14     74  2  37    10  3  39     14  2  21 


WINE  MEASURE. 

tuns  lihd.  gal.  qt.  pt.  tuns,  hhd.  gal.  qt.  pt, 

Ex.  1. 456     2     24      1     0  2.  257     3     10     1      I 

399     3     46     3     1  199     0     50     3     1 


tuns,  hhd.  gai.  qt.  pt,  tuns,  hhd.  gal. 

3. 467     2       0     0     0  4.  27     2     54 

299     3     32     2     I  19     3     62 


hhd.  gal.  qt.         hhd.  gal.  qt.      gal.  qt.  pt. 
5.  147     14     2     6.  14        1     2     7.  24     2     0 
79       3     3  12     41      3  17     0     1 


DRY  MEASURE. 

bu.  pks.  gal.     bu.  pks.  gal.     bu.  pks.  gal 
1.  86     3      i     2.  59     1     0     3.  ()2     0     0 
46     1     0         39     3      I         24     1      1 


COMPOUND    SUBTRACTION.  71 

pks.  gal.  qts.   pks.  gal.qts.  pks.  gal.  qts.  pts. 
Ex.  4.  67  0  2  5.  28  0  I   6.  74  1   1   I 
32  1   1     12  1  3    27  I  3  1 


TIME. 


cL  hr.  mim.   d.  hr.  min.  sec.   mo.  w.  d,  hr. 

Ex.  1.  37  2  39  2.74  3  12  14  S.  46  I   1   4 

29  21  49  "   47  21  54  36  •   29  3  6  Z\ 


w,  d.  hr.   m.  s.   ysr.  m.  w.    m.  w.  d. 

4.  36  0   0   0   0  5.  17   10  2  6.147  2  3 

35     6  23  50  59    14  12  3     19  2  4 


d.  hrs.  111.       hrs.  min.  sec. 
7.  167  21  50      8.  174  50  51 
19  23  54         94  59  57 


MISCELLANEOUS  EXAMPLES  IN  SUBTRACTION. 

Ex,  1.  I  borrowed  of  a  friend  five  hundred  guineas, 
and  hav^  paid  at  different  tim.es,  three  hundred  and 
ninety  pounds  six  shillings  and  seven  pence  three  far- 
things :  what  have  I  still  to  pay  ? 

Answer,   134^.  ISs.  44^. 

2.  A  horse  and  his  harness  are  worth  175  dol.,  but  the 
harness  is  worth  47  I).  37^  cts.  I  demand  the  value  of 
the  horse  ?  Ans.  127  D.  623^  cts. 

3.  What  sum  added  to  150  guineas,  will  make  up 
1 99^  9.S.  9^  ?  Ans.  41  /.  I9s.  9ld. 

4.  At  an  eclipse  of  the  sun,  the  moon  is  situated  be- 
tween the  earth  and  sun  ;  how  far  distant  is  the  moon 
from  the  sun,  supposing  the  distance  between  the  earth 
and  the  sun  95  millions  of  miles,  and  that  between  the 
earth  and  moon  240  thousand  ? 

Ans.  94760000  miles. 


7*2  COMPOUND    SUBTRACTIOK. 

5.  The  great  bell  at  Oxford  weighs  7  tons,  1 1  cwt, 
3qrs.  41b.  ;  that  at  St.  Paul's  5  tons,  2  cwt.  Iqr.  22lb.; 
and  the  great  Tom  of  Lincoln  weighs  4  tons.  16  cwt. 
3  q.rs.  16  lb.  :  how  much  heavier  than  these  together  is 
the  great  bell  at  Moscow,  .which  is  198  tons  ? 

A'iS>  180  tons,  8  c»vi.  3  qrs.  14-  lb. 

6.  The  Royal  Exchange  cost  80  thousand  pounds  in 
building; ;  the  Mansion-house  40  thousand  ;  Blackfnars- 
bridyje,  153  thousand;  Westminster-bridge,  389  thou- 
sand ;  and  the  Monument,  13  thousand!  pounds;  but  the 
Cathedral  of  8t.  Paul's  cost  »00  thousand  :  how  much 
did  this  cost  more  than  all  the  resr  ?       Ans.  125000^. 

7.  If  my  income  is  3G7/.  8.s  A-jd.  and  my  expenditure 
be  340  guineas  :  how  much  can  1  lay  by  ? 

Ans.  10/  8s   4]i. 

8.  A  person,  by  great  losses,  was  oblij^ed  to  call  his 
creditors  together  :  lie  found  his  whole  property  amount 
to  .'527/.  12s.  8ld.  ;  but  he  owed  to  one  man  150/.  ;  to 
another  300  guineas  ;  to  a  third  20  crowns;  to  a  fourth 
55/.  8s.  9ld.  ;  and  to  a  fifth  200  guineas  ;  how  much  will 
they  be  losers  ?  Ans.  207/.  1 6s.  Old. 

9.  A  gentleman  leaves  between  his  two  children  50,000 
dollars  ;  to  the  younger  he  leaves  17478  dollars  :  what 
was  the  fortune  of  the  elder  ?•        Ans.  32522  Dollars. 

10.  An  apprentice  has  served  of  his  term  of  seven 
years,  three  years,  two  months,  three  weeks,  four  days, 
seventeen  hours  ;  how  much  longer  has  he  to  serve  ? 

Ans.  3  yrs.  10  m.  0  w.  2  da.  7  ho. 

1 1.  From  a  field  of  6|  acres,  I  take  out  two  gardens, 
one  measuring  ia  roods,  and  the  other  24  roods,  and  a 
piece  of  ground  for  coach-house  and  stables,  that  mea- 
sures 1  rood  and  12  perches  :  what  will  be  the  size  of 
tht  field  after  these  pieces  are  taken  away  .? 

Ans.  4  ac.  1  r.  38  poles. 

12.  A  plumber  puts  lead  upon  the  diiTerent  parts  of 
my  house  that  weighs  5  cwt.  3  qr.  ;  and  he  takes  avuy, 
in  return,  old  lead  weighing  2  cwt.  24  lb.  :  what  is  the 
difterence  in  the  vreight  between  the  new  an'i  the  old 
lead  ?  Ans.  3  cwt.  2  qrs.  4  lb. 


(") 


COMPOUND  MULTIPLICATION 


Is  the  method  of  finding  the  amount  of  any  given  num- 
ber of  different  denominations,  by  repeating  it  any  num- 
ber of  times : 

I.  When  the  given  multiplier  does  not  exceed  13. 

RuLK.  Write  the  multiplier  under  the  lowest  denomi- 
nation of  the  multiplicand,  multiply  every  number  of  the 
multiplicand  by  the  multiplier,  and  bring  the  several 
products  as  they  occur,  to  the  next  higher  denomination. 
Write  down  the  remainders,  and  carry  the  integers  to 
the  next  product. 

Ex.  Multiply  Z..768  14s.  9\d,  by  9. 


Ex. 


Dlls.  cts. 

L. 

768 

s. 
14 

H 

9 

4 

Dlls 

.cts 

691B 

13 

i! 

] 

Ex.    1.  79    14 
Ex.  3.  67  37| 

X3 
X4 

Ex.2. 
Ex.4. 

84 
79 

62| 
25 

X6 
X5 

Ea, 

.  D.  d.  cts 

}. 

Ea.D. 

d. 

cts.  mis. 

5.  7 

7     5     4X7 

Ex. 

6.  6     3 

4 

7 

6X9 

Ea.  D.  cts.  Ea.  D.  cts. 

Ex.  7.  74     3     50  X  8     Ex.  8.  29    4    43^  X  12 


?4  COMPOUND    MULTIPLICATION. 

L.    s.    d.  L.    s.      d, 

Ex.  9.  3987  4  6|  X  2 
11,  2987  3  95  >^  5 
13.  3487  12  8  X  6 
15.  5094  10  11^  X  8 
17.  •37(-4.  12  h\  X  10 
19.  4610  15  4  X  12 
21.  1456  16  10  X  12 
23.  3420  13  5*  X  10 
25  2675  19  31  X  9 
27.  467i)  17  8*  X  1 1 
II.  VMien  the  multiplier  is  a  composite  number,  and 
can  he  resolved  into  tuo  or  more  component  parts. 

Rule.  JMultipIv  bv  its  component  parts  successively, 
and  the  last  product  will  be  the  answer. 

Ex.  Multiply  L.374  10s.  ll^c?.  by  63. 
L.     s,    d. 

374  10  11^  X  63ZI9   X  7 
9 


10.  3564 

10 

n  X 

3 

12.  2648 

16 

81  X 

5 

14.3498 

2 

6i  X 

7 

16.  2691 

18 

lU  X 

9 

18.  3465 

15 

10^  X 

11 

20.  359  1 

19 

9^  X 

4 

22.2761 

14 

4  X 

6 

24.  4694 

12 

7  X 

8 

26. 3476 

17 

81  X 

5 

28.  4900 

0 

92  X 

7 

3370  1 

8  9l 
7 

Ans. 

23596  ] 

[1  8* 

EXAMPLES. 

L 

s. 

d. 

L. 

s. 

d. 

Ex.  1. 

456 

12 

%   X 

15 

Ex.  2. 

436 

14 

31 

X 

16 

3. 

784 

15 

4  X 

18 

4. 

397 

16 

10 

X 

21 

5. 

674 

18 

lOl  X 

22 

6. 

487 

16 

9^ 

X 

24 

7. 

245 

10 

3  X 

30 

8. 

376 

15 

11 

X 

30 

9. 

246 

19 

91  X 

35 

10 

489 

18 

81 

X 

42 

11. 

397 

13 

3  X 

48 

12- 

369 

10 

2 

X 

54 

13. 

384 

15 

10*  X 

56 

14. 

565 

i2 

91 

X 

63 

15 

592 

12 

9  X 

66 

16. 

80U 

9 

8 

X 

72 

17. 

911 

13 

21  X 

84 

18 

914 

16 

4 

X 

77 

19. 

397 

4 

41  X 

96 

20. 

S74 

12 

51 

X 

108 

21 

459 

9 

91  X 

100 

22. 

279 

13 

3 

X 

120 

23 

S76 

15 

4  X 

121 

24 

347 

3 

9 

X 

132 

25. 

376 

4 

91  X 

144 

26. 

5b7 

14 

7 

X 

45 

27. 

897 

16 

0  X 

108 

28. 

675 

13 

31 

X 

88 

29. 

487 

19 

111  X 

121 

30. 

hb<^ 

12 

2 

X 

132 

COMPOUND    MULTIPLICATIOST.  75 

III.  When  the  multiplier  is  not  a  composite  number. 

Rule.  Take  the  composite  number  which  is  nearest 
to  It,  and  multiply  by  the  component  parts,  as  before  : 
then  add  or  subtract  as  many  times  the  first  line,  as  the 
composite  number  is  less  or  greater  than  the  given  mul- 
tiplier. 

(1)  Multiply  L.324  12*-.  6^  by  394. 
L.     s.     d 
3^4   12     6*   X   394  IZ  8  X  7  X  7  +  2. 
8 


The  nearest  composite  number  is 
392  =8X7X7;  I  accordingly 
multiply  by  these  three  figures,  and 
to  the  product  1  add  twice  the  ori- 
ginal sum,  which  gives  the  true 
answer. 


EXAMPLES. 


2597     0 

18179     2 

649     5 

127903     1     5 


L    s. 

d. 

L.     s. 

d. 

Ex.  1.574  12 

6*   X 

38 

Ex.  2.  387    18 

71  X 

46 

3.  325     8 

4     X 

58 

4.' 222   12 

8^   X 

68 

5.  2  d   18 

91  X 

78 

6    136   14 

5     X 

94 

7.  300     0 

3*    X 

273 

8.  246    12 

0^   X 

359 

9.  S'-IS   16 

Ol  X 

412 

10.  326   18 

3     X 

687 

11.239     9 

9     X 

740 

12.  560     0 

^21   X 

388 

13.  660  15 

4,1   X 

1004 

14.  407    13 

1      X 

1325 

15.700     0     01   X    1450  16.110  10  11     X   1208 

EXAMPLES  OF   IVEIGHfS  AND  MEASURES, 

TROY  WEIGHT. 

lb.  oz.  dwt.  gr.  lb.  oz.  dwt.gr. 

Ex.  1.    187  9      12  20    X      4  Ex.  2.  256     6     0  252  X  5 

3.  169   6      14   17   X      6         4.  J79  1 1      9     y  X  7 

5.  254  3       3     3X9          6.  253  11      4  20  X  8 

7.  675  4      15  10  X    11          8.   375  0     0  17  X  12 


76  COMPOUND   MULTIPLICATrON. 

AVOIRDUPOIS  WEIGHT. 

tons.cwt.qr.lb.  oz.  dr.  tons,cwt.qr.  lb.  oz.dr. 

i5xJ.l2  10  3  14  10  12X2  Ex.  2,  6i  13  2  15  6  8X4 
3.25  0  2  8  4  4X3  4.46  15  5  12  4  4X6 
5.75  13  0  18     6   10X8        6.  39    12  2  16   10  8X9 

APOTHECARIES'   WEIGHT. 

lb.  oz.  dr.  sc.  lb.  oz.  dr.  sc. 

Ex.   1.  456     3     4     1    X     5  Ex.  2.  748     5     2  2  X    8 
3.  534     7     6     2   X    12  4.  378   10     0  1  Xll 

5.  321     5     4     IX   10  6.  491     5     7  2x9 

CI.OTH  MEASURE. 
yds.  qr.  nl.  E.e.  qr.  nl.  yds.  qr.  nl. 

Ex.  1.  210  2  1  X  4     2.  378  4  3  X  7     3.  596  3   1  X  12 
4.  357  1  3  X  6     5.  738  3  2  X  9     6.  876  0  3  X  10 
LONG    MEASURE. 

yds,  ft.    in.  be.  lea. 

Ex.  1.556     2     10     1X5    Ex.  2.  379 

3.  369      1        9     2X8  4.  376 

5.  241     2     11      1  X  10  6.674 

LAND  MEASURE. 


m. 

1 
2 
2 

fur.    p. 

6  20  X  7 
5     37  X  9 

7  18X  6 

r. 

3 
2 
0 

P- 

12   X   12 
25   X    10 
12  X     8 

ac.  r.  p.  ac. 

Ex.  1.  456  O  25  X   11  Ex.  2.  597 

3.  371  2  18  X     4  4.  271 

5.    189  3  32  X     8  6.  430 

LIQUID  MEASURE. 

tuns,hhd.  gal.  qts.  p.  tuns,  hhd.gal.qts. 

Ex.  1.456  3  28  2  1x4  2.456  3  46  2x6 
3.374  2  60  3  1X8  4.350  2  25  1X2 
5.  221      1        4     1     0X5     6.124     3     50     3X10 

DRY   MEASURE. 

bu.  pks.  gal.  bu.  pks.  gal. 

Ex.  1.29     2     1X3  Ex.  2.  29     3      1    X  5 

3.  76     3     0   X   4  4.  27     2     1    X   6 

pks.  gal.  qts.  pts.  bu.  pks.gal.qts. 

5.  r.4      I      3      1    X  7  6.  64     2     1     2  X     8 

7.  76      1     2     0   X     9  8-37      1      13x11 

9.  6^     0     3     1   X   12  10. 64     3     1     2   X    12 


COMPOUND    MULTIPLICATION  77 

TIME. 

w.  d.  hrs-m.  s.  yrs.  mo.  w.d.' 

Ex.  1.  73  6  10  40  30  X  5  Ex.  S.  594  12  3  4  X  7 
3.  36  4  1-2  15  20  X  9  4.  364  8  2  6  X  8 
5.  98  5   17   13  53   X  iZ  6.  443  lO  3  3  X  11 


MISCELLANEOUS    EXAMPLES. 

Ex.  1.  What  cost  12  lb.  of  tea  at  1  dol-  50  cts.  per  lb.  ? 

Ansv\'er.   IS.'dolls. 

2.  What  cost  16^  lb.  of  suj^ar,  at  Is,  \ld.  per.  lb-  ? 

Ans.  1  «.s.  nld. 

3.  What  is  the  value  of  24  yards  of  Irish  linen,  at  3s. 
6^.  per  yard  ?  Ans,  4/.  5s. 

4.  What  will  79  bibles  come  to,  at  Idol.  12^  cts. 
each?  Ans.  88  dol.  87^  cts. 

5.  What  is  the  value  of  85  gallons  of  brandy,  at  l9s. 
Old.  per  gallon  ?  Ans.  84^.  2s.  3ld. 

6.  What  is  the  weight  of  28  ingots  of  gold,  each  weigh- 
ing 6  lb.  7  oz.  15  dwts.  20  gr.  s' 

Ans.  I  86  lb.  2  oz.  3  dwt.  8  gr. 

7.  What  will  157  oxen  cost  at  15/.  5s.  9^.  each  ? 

Ans.  2400/.  2.S-.  9d. 

8.  What  is  the  value  of  576  sheep,  at  1/.  6s.  sU.  each  ? 

'  Ans.  756/   Oa.Od, 

9.  How  much  must  I  pay  for  759  chaldrons  of  coals, 
at  58s.  6rf.  per  chaldron  ?  Ans.  2220/.  Is.  6d. 

10.  What  is  the  value  of  199  firkins  of  ale,  at  I2s  6d. 
per  firkin  :'  Ans.  124/.  7s.  6d. 

11.  What  is  the  value  of  245  yards  of  broad  cloth,  at 
19s.  Id.  per  yard  ?  Ans.  239/  17.^.  l\d. 

12.  What  is  the  worth  of  a  stack  of  hay,  containing 
75  loads,  at  3/.  19s.  9d.  per  load  ?     Ans.   299/.  Is.  3 J. 

13.  What  is  the  worth  of  12^  lb.  of  cofiee,  at  25  cts. 
per  lb  ?  Ans.  3  dol.  l-ig  cts. 

14.  How  many  pounds  sterling  are  there  in  28  purses, 
each  containing  15  guineas,  15  half-guineas,  15  seven- 
shilling  pieces,  and  tliree  crowns  ?     Ans.  829/.  10s.  Od, 

15.  What  is  the  weight  of  1 000  guineas,  each  guinea 
weighing  5  dwts.  9|  gr.  ?    Ans.  22  lb.  3  oz.  15  dwt.  20  gr. 

7* 


78 


COMPOUND    MULTIPLICATION, 


16.  I  bought  at  a  sale  47a  dozen  of  port  wine,  at  2l.  5s, 
6d.  per  dozen,  how  much  money  must  1  send  to  pay  for 
it?  Ans.  108/.  Is.  3d. 

17.  What  is  the  value  of  83  tons  of  iron  at  18^.  17s. 
^^.  per  ton  ?  Ans.  1605/.  12s.  3^. 

18.  What  do  79  packages  of  good*  weigh,  supposing 
that  each  package  weighs  3  cwt,  3  qrs.  15  lb.  ? 

Ans.  15  tons.  6  cwt.  3  qr.  9  lb. 

19.  If  one  ounce  of  gold  cost  3/.  16s.  8(/ ,  what  is  the 
value  of  43C4  ounces  ?  Ans.  1673^.  5h.  Od. 

20.  What  shall  I  pay  annually  for  459  acres  of  land, 
at  2  dol   37^  cts.  per  acre  ?  Ans.  1090  dol.  12J  cts. 

21.  What  is  the  price  of  185  gallons  of  rum,  at  l3s. 
&ld   per  gal.  }  Ans.  125/.  5s.  2*. 

22.  If  a  man  spend  1  dol.  62a  cts.  per  day,  how  much 
does  he  expend  in  a  year  ?  Ans.  593  dol.  I2a  cts. 

23.  How  much  federal  money  in  49/.  sterling,  allow- 
ing 4  dol.  44  cts.  to  a  pound  sterling  ? 

Ans.  217  dol.  56  cts. 


BILLS  OF  PARCELS. 
A  mercer's  bill. 

L  s.  d.  L.  s, 

12  yards  of  silk,  at        -        0  15  ?  peryard| 
114  Do.     of  flowered  silk  at  0  18  7^ 
of  velvet,  at       -      1     2  4 
of  satin,  at         -     0  13  9 
of  brocade,  at     -    0  15  7 
of  lustring,  at     -    0     6  3 


16  Do. 
12  Do. 
27  Do. 
14  Do. 


I      I 


A  stationer's  bill. 

L 

250  Reams  of  paper,  at  -  1 

112  D».    do.     at  -  2 

34  Do  of  imperial  brown  at  1 

500  Dutch  quills,  at       -       0 

2500  Do.    common,  at    -     0 


s.d                 L. 

s. 

2  6  per  ream 

4  6 

15  0 

3  9  per  hun. 

2  3 

d. 


OOMPOUND    MULTIPLIOATION. 


79 


A  carpenter's  bill. 


65  cubick  feet  of  oak,  at 
125  Do.  wrou;^lit  and  framed,  at 
176  Do.  fir  frained  aad  mould- 
ed, at         -         -         - 
15  square  shed  roofing,  at 
8   Do.  hip  aad  valley  roof- 


70  feet  water  trunk,  at       -       € 
3C>4  feet  ovolo  wainscot  sashes, 

at         -        -         -    0 
124  Do.  do.  mahogany,  at         1 

10  men's  labour,  for  25  days,  at  4 


d. 

3  per  foot 
8 

L. 

s. 

6 

6  per  square 

3 
lo  per  foot 

9    - 

4 

8  per  day 

1 

A    BRICKLAYER  S    BILL. 


39  rod  of  grey -stock  brick- 
work, at 
7   Do.  in  party  wall,  at 
105  feet  of  J  8  inch  drain,  at 
1050  Do.  of  pointing  old  work 

at  -  -  - 
1500  grey  stocks,  at 
125  pan-tiles,  at 
45  hods  of  mortar,  at 
13  Do.  of  tarras,  at  -  - 
15  bricklayers,  25  days,  at  0 
12  labourers,  ditto,  at  -  0 
66  load  of  rubbish  carted 

away,  at  0 


L.  s.  d. 


L.  s.  d. 


13  0  per  rod 
15  0 

3  0  per  foot 

(\5l        - 

4  6  per  bun. 
0  1^  each 

0  7 

4  2 

4  6  per  day 

3  0 

2  6prload. 


ao 


COMPOUND    MULTIPLICATION. 


A  slater's  bill. 
L.  s.  d. 
9  square  of  Westmore- 
land slatinu;,  at     -     2  19  6  per  square 
7  do.  of  Welsh  ladies,  at  1   IT  4  - 

5  Do.  of  Welsh  coun- 
tess, at  1  18  3 
35  Do.  of  ripped  and  rub- 


L.  s.  d. 


bish  cleared,  at        0 

2 

7 

12  slaters  7  days,  at       0 

4 

5  per  day 

tJ  labourers,  do.  at          0 

2 

9 

5050  clout  nails,  at              0 

0 

4  per  hun. 

painter's  adn 

glazier's  bill 

J 

*• 
1035  yards  of  painting  3  times 

u. 

in  oil,  at 

0 

7^  per  yard 

56.5  Do.  do.  and  sand,  at 

1 

3 

3(5  sash  frames,  at 

0 

11  each 

432  sash  squares,  at 

0 

8a  per  doz. 

12()5  feet  of  best  Newcastle 

j^lass,  at 

1 

71  per  foot 

356  Do.   Iar«;e  size,  at 

2 

1.^ 

iOOO  Do.  in  lead  work,  at 

1 

05 

L.  s.  d. 


COMPOUND  DIVISION, 

Is  the  method  of  finding  how  often  one  ^iven  number 
is  contained  in  another  of  different  denominations;  or, 
to  divide  a  given  compound  number  into  any  proposed 
number  of  equal  parts. 

I.  When  the  given  divisor  does  not  exceed  12. 

Rule.  Place  the  divisor  to  tlie  left-hand  of  the  divi- 
dend. Divide  the  highest  denomination  of  the  dividend 
by  the  divisor,  and  write  down  the  quotient ;  reduce 
the  remainder,  if  any,  into  the  next  lowerdenomination, 
adding  to  it  the  aumber  which  stands  in  that  place  oC 


COMPOUND    DIVISION.  6l 

the  dividend,  and  divide  as  before,  and  so  proceed  to 
the  end. 

Ex.  1695?.  14s.  4ld.  H-  8. 
L.      s.     d. 

8)1695   14     4^ 


211 

19 

3^—2 

• 

8 

Proof  1695 

14 

^l 

EXAMPLES. 

D.  cts. 

D.  cts; 

Ex.  1.  74  50    -T-  S 

Ex.  2,  62  25    -i-  4 

3.  56  371  -^  5 

4.  79   18i  -T-  6 

5.  49  49    ^7 

6.  63  25    -T-  8 

Ea.  I),  cts. 

Ea.  D.cts. 

7.  43  7  37^  -7-    9 

8.  56  6  25  -r-  10 

9.   17  4  50    -^  n 

10.  13  7  75  H-  12 

L.      s,     d. 

L.     s.    rf. 

Ex.tl. 

457     8     9^  ^    3 

Ex 

.  12.579   18     4*H-     2 

13. 

3;^6  18     7i  -=-    4 

' 

14.  768     2     6^  -i-    5 

15. 

474    12   10  -r-    6 

I6. 93t  14     5  H-     7 

17. 

897    16     4  -j-    8 

18.  2.56    17    10^  -r-  10 

19. 

759     0     0-^9 

20  694   19     6-7-12 

21. 

101    15     9l~-  1( 

22.496     0     0    -i-  12 

23. 

900     0     0-^8 

24.  500     5     5    -f-    4 

25. 

800   10     2    -i-     7 

26.  270   17     7|  -T-     6 

2r. 

464     3     91  -V-    6 

28.  901      1      1     -r-     9 

II.  When  the  divisor  is  a  composite  numher. 

Rule.    Divide   bj  the  component  parts  of  the  divisor 
successively,  and  the  last  quotient  will  be  the  answer* 
Ex.  Xl48  8s.  8^.  -7-  27  =  3  -^  9. 

L.     s.     d. 

3)148     8     8^ 


9H9     9     6i— 


61-n 

111—6  J 


5     9  '^ 

The  answer  is  5/.  9s.  1 1]^'  ^^. 


82 


COMPOUND    DIVISION. 


;x. 


L. 

s. 

d. 

L. 

s. 

(i. 

1    167 

12 

Gl^ 

_ 

14 

E5 

:.  2.  769 

9 

81- 

-     20 

3.  33^ 

15 

8^ 

- 

15 

4.  594 

7 

6   - 

-     25 

5.  4.86 

9 

9   - 

- 

16 

6.  333 

10 

lOi- 

-     28 

T.^67 

0 

ol- 

- 

18 

8.  498 

9 

9^- 

-     32 

9.  4:<9 

5 

61 -i 

- 

24 

10.  596 

12 

ri- 

-     36 

11.  37^ 

18 

7    - 

- 

27 

12.  465 

11 

11  - 

-     44 

13    487 

9 

9^- 

•- 

30 

14.  564 

13 

5^ 

-      49 

15.  596 

4 

6  - 

- 

33 

16.   678 

6 

5  - 

7-     54 

17.834 

3 

6^H 

- 

42 

18.  999 

9 

8   - 

r-     56 

19. 32: 

U 

4    H 

- 

48 

20.  564 

4 

6  - 

-     63 

21    387 

12 

11    H 

- 

72 

22.  248 

3 

0  - 

-     84 

23.  565 

II 

8   - 

- 

88 

24.'  505 

5 

51- 

-     99 

25.  674 

18 

8H 

- 

108 

26.  564 

2 

2  - 

-    i20 

27.465 

3 

3  ^ 

- 

132 

28.  888 

8 

8   - 

-    144 

When  there  are 

three  component  parts. 

Ex.  L.  1350  10s. 

llrf 

.-r- 

240  =5 

X6 

X  8 

, 

L. 

s. 

d. 

5)1350 

10 

u 

6)270 

2 

2- 

—  4 

r 

8) 

45 

0 

4^ 

~n.- 

i  Air 

Ex.  1.L.5527  10s.  6^.-J-243.  2.  18568/.  12s.  1^^-4-1296. 

III.  When  the  divisor  is  greater  than  12,  and  not  a 
composite  number  ? 

Rule.  The  several  quotients  must  be  found  by  the 
method  of  Lon^  Division,  (see  pp.  28  and  29),  reducing 
the  remainders  to  the  next  lower  denomination,  and  tak- 
ing in  tho!^e  numbers  of  the  dividend  which  are  of  the 
same  denomination. 


COMPOUND    DIVISION. 


83 


Ex.  Divide  L.l 350  lOs.  lie?,  bv  242. 
L.       s.     d. 
242) J 350   10  11(5 
1210 

140 
20 


242)2810(1 
2662 


148 
12 


242)1787(7- 
16'J4 


93 
4 


242)37:^(1 
242 


D.  cts. 

Ex.  I.  234  50  -4- 

3.  427  C)2l  — 

Ea.  I),  cts, 

5.   17  3  8  ^ 

7.  r23  4  J5 

L.     s.     d. 


ISO 

17 
37 


9.  985  18 

11  405  16  4^-7- 

13.  565  13  3- 

15.  800  8 

17.  987  14 

19.  598  12 

21.  483  6 

23.  98ci  5 

25.  U«5  19 

27.  2690  12 


9   -7- 


8^ 
4 
6 
6 

2 
3 


■59 

-  74 

19 

29 
37 
41 
46 
67 
73 
89 
107 
166 


^  D.  cts. 
Ex.  2.*  627  25  -J-  26 
4.  317  75  ^  43 
Ea.  D.  ct-. 


6.  127  7   12^ 


68 


8.  319  4  50  -T-  89 

/..  s.     d, 

10.  IO(M  12  111 

12.  2468  13  Zl 

14.  .5746  9  6  -T- 
16.  6321   3   3  -r- 

18.  4'^68  12  8 

20.  4«21  9  71 


22.  5943  16  6 

24.  3618  4  6  — 

26.  4683  15  5^,  -^  376 


28.  5649  9  9 


23 
39 
59 
61 

69 
87 
97 
97 


439 


84  gOMPOUND    DIVISION. 


L.    s,    d. 

L.      s.    d. 

29. 

6259  11     6  H-     215 

30.  3604  10     0  -T-    509 

Si. 

P654     7     71-7-     649 

32.  6534   16     3^  -^    606 

S3. 

5942   17     ^\~    757 

34.  4.593  12     4  -^  1585 

35. 

4628     .5     9-7-  1001 

36.  5349     0     0-7-4786 

37. 

1456   16     7   -T-  3761 

38.  9504     1      1*  -=-  8078 

IV, 

When  the  divisor  consists 

of  a  numher  not  exceed- 

ing  12,  with  one  or  more  cyphers. 
Rule.  Cut  off,  by  a  line,  as  many  places  in  the  pounds 
as  there  are  cyphers  in  the  divisor,  and   divide  by  short 
division  ;  then  i educe  the  remainder  to  the  next   lower 
denomination,  as  in  the  last  rule. 

Ex.  Divide  L  5645   14s.  4d.  by  1200. 
12.00-56  45    14     4 


L. 

4  —  845 
20 

12.00;  169. 14 

s.  14— 114 
12 

12.00J13.72 

d.  1—172 

EXAMPLES  OF  WEIGHTS  AND  MEASURES. 

TROY  WEIGHT. 

lb.  oz.dwt;.  gr.  lb.  oz.dwt.gr. 

Ex.  1.  287     9    12  20  -T-     4  Ex.  2.  356     6  0  22  -^  5 

3.269     6  14     7  -=-     6          4.  379  11   9     0 -r-  7 

5.  854     3     3     3  -r-     9           6    3j5   11   4  20  -^  8 

7.  675     4   15   10  -f-   II           8.  775     0  0   17  -i-  12 

AVOIRDUPOIb  Wti:iGHT. 

tons,  cwt.  qr.  Ih.  oz.  dr.  ton*,  cvvt.qr.lb.  oz.  dr. 

1,  412  10  3  14  10  12  -^  2  2.  664  13  1  12  6  8  -r-  4 
3.  526  0  0  18  6  6 -~  3  4  464  0  3  27  0  S  -~  Q 
5.678     2  2     2     8     2  ~- 8     6.591     5  0     4-3   12 -i- 9 


COMPOUND    DIVISION. 


83 


APOTHECARIES  WEIGHT. 

lb.  oz.  dr.  scr.  lb.  oz.  dr.  ser. 

Er.   1.  591    8     4     1    -~  5Ex.  2.  748    5     7     0  -=-    8 

3.  639    1      1     2  H-  12          4.  392  10     6     0  ~  II 

5.  487   2     0     0  -7-  10          6.  421     4     5     1-7-9 

CLOTH  MEASURE. 

yds.  qr.  n.  E.e.  qr.  n. 

Ex.    1.  5210    2     1-5-4  Ex.  2.  596*     3  I  -r-  1 1 

3.   3976     1     2  -i-  6  4.  7645     4  2-^12 

5.   4721    0     0  -r-  8  6.  3492     0  3  -r-    9 

LONG    MEASURE. 

yds.  ft.  in.  b.c.                    lea.  m.  fur.   p. 

Ex.  1.  5946    2    10    1  -T-     5  Ex.  2.  3795  2     7     30-7-7 

3.4736     1      8    2  H-     8           4-4965  1      3      18  -f- 0 

5.2005  0   11    2  H-  10          6.6743  2     6       4 -i- 6 

LAND  MEASURE. 


ac.  r.  p. 

Ex.  1.    654  2  24-7-11 

3.    371  0  18  -H    4 

5.     891  3  32-7-     8 


ac.  r.  p. 

Ex.2.  958  3  12-7-12 

4.  379  0  25-7-10 

6.  496  1  1-^8 


Ex.  1. 

2. 

3. 
4. 
5. 
6. 


LIQUID  MEASURE. 

tuns,  hbd.  gai.  qts.  pt. 

456  3  27  2  1   -t- 

656  3  31  2  0  -r- 

594  0  30  3  0  -j- 

391  2  25  1  0-7- 

271  0         O  2  0-7- 

421  3  50  3  0  -T-    10 

DRY  MEASURE. 


bu.  pks.  gjal. 
Ex.  2.  87     3     1-7-5 


bu.  pks.gal. 
Ex.  1.  16     2     1-7-3 

pks.  gal.  qts.  pts.  pks.  gal   qts.  pts. 

Ex.  3.  327    1       3     0 -i-     7  Ex.  4.  219  ^0     2     0  -r-     9 
5.  12.)  0      2      1-7-11  6.    99      1     3     0-7-12 

8 


86  COMPOUND    DIVISION. 

TIME. 

w.  d.  hrs.m.  sec.  yrs.  mo.  w.d. 

Ex.  1.  779  6  20  40  25  -4-    5  Ex.  2.  594  12  2  4  -f-    7 
3.  S91   4  12  16  12 -i-    9  4.954     6  3  5-4-6 

5.913  0     4     0     5-4-J2  6.348   10  3  3-^11 


MISCELLANEOUS  EXAMPLES. 

Ex.  1.  If  17  yards  of  doth  cost  19/.  3s.  9rf.,  what  is  it 
per  yard  ?  Answer.  I/.  2s,  tiliL-^-^. 

2.  V\  hat  is  the  price  of  one  pound  of  sugar,  if  8Jb.  cost 
nine  shillings  ?  Ans.  Is.  lid. 

3.  I  he  expenses  of  a  journey  amounting  to  97/.'9s.  6rf. 
are  to  be  defrayed  by  six  persons  :  how  much  will  each 
have  topay  ?  Ans,  16/.  4*-.  lid. 

4.  I  have  bought  12  gallons  of  wine  for  32  dollars 
50  cts. ;  how  much  is  that  per  gallon  ? 

Ans.  2  dolls.  70  cts 

5.  Twelve  boys  are  to  have  a  guinea  and  a  half  divi- 
ded among  them  :  what  will  be  each  boy's  share  ? 

Ans.  2s.  7ld. 

6.  A  hundred  and  twenty -five  sailors  have  taken  8465/. 
prize  money  :  how  much  will  each  man  be  entitled  to  ? 

Ans.  67/.  U>.4,ld,  ^. 

7.  I  have  bought  144  pair  of  stockings  for  27/. ;  at  what 
rate  can  1  sell  them  so  as  to  gain  by  each  pair  one  shil- 
ling .►*  Ans.  4s.  yd, 

8.  What  did  I  pay  a  piece  for  sheep,  having  bought 
75  for  135/.  r 

9.  Cheese  at  3/.  12s.  dd,  per  cwt.  : 
per  1 .'.  ? 

10.  If  81  oxen  cost  1781/.  12s.  6d, : 
of  one  }  An.%.  2 1 

11.  Ifapipeof  wine  cost  95/. :  how 
dozen,  which  contains  three  gallons  ? 

Ans. 

12.  Bought  50  dozen  of  wine  for  a  hundretl  guineas  : 
how  much  is  thai  per  bottle  ?  Ans.  3s.  (jd, 

13.  Divide  a  tliousand  guineas  bitweei.  5i3  |jeoj)h',  and 
see  how  much  it  is  tor  each  ."^      Ans.  4d/.  13s.  O^d,  4* 


Ans. 

1/. 

16s. 

how  much 

is  that 

Ans. 

7ld. 

8 

iia» 

w  hat  is 

,  tht 

value 

/.  19s. 

Wld.  I 

/  much 

is 

that  a 

.  2L  5s,  2rf. 

108 
136' 

MISCELLANEOUS  QUESTIONS.  87 

14.  If  12  pieces  of  linen  cloth  contain  250  yards,  what 
is  the  length  of  a  single  piece  ? 

Ans.  20  yds.  3  qr.  1*9  nail. 

15.  How  much  can  I  afford  to  spend  a  day,  a  week, 
and  a  month,  if  my  income  Ire  5(iO/.  per  annum,  allow- 
ing 52  weeks,  or  13  months  to  a  year  ? 

Ans.     1/.     7.S-.  4>ld.  per  day. 
9/.   12s.  3^.  per  week. 
•  S8/.     9s.  2ld.   per  month. 

16.  If  12  tea-spoons  weigh  9  oz.  17  dwt.  12  gr.  ;  what 
is  the  weight  of  each  spoon  ?         Ans.  16  dwts.  1 1  gr. 

MISCELLANEOUS    qUESTIONS. 

Ex.  1 .  It  is  said  that  Syrius,  or  the  Dog  Star,  is  the 
nearest  of  all  the  fixed  stars,  and  that  its  distance  is  com- 
puted at  2,200.000,000,000  miles ;  how  many  years,  (each 
containing  365  days,  6  hours  exactly,)  would  a  cannon 
ball  be  in  passing  from  the  earth  to  8irius,  supposing  it 
travelled  at  the  rate  of  480  miles  per  hour  ? 

Ans.  522853S. 

Ex.  2.  The  Planet  Mercury  is  about  thirty -seven  mil- 
lions of  miles  from  the  Sun  ;  Venus  sixty-eight  millions  ; 
the  Earth  ninety-five  millions  ;  Mars  a  hundred  and 
forty  five  millions  ;  Jupiter  four  hundred  and  ninety- 
three  millions  ;  Saturn  nine  hundred  and  eight,  and  the 
Herschel  one  thousand  eight  hundred  millions  of  miles 
from  the  Sun  :  put  these  several  distances  down  in 
figures,  and  add  them  together  as  a  sum  in  Addition. 

Ans.  3546.000.000 

Ex.  3.  How  much  nearer  the  Sun,  is  Mercury  than 
Mars;  and  how  much  farther  is  the  Herschel  than  the 
Earth  ?  See  Ex.  2.  Ans.  Mercury  108  millions  nearer  the 
sun  than  Mars,  and  Herschel  ^705  millions  further  from 
the  Sun  than  the  Earth. 

Ex.  4.  The  beautiful  planet  Venus  travels,  in  her  an- 
nual journey  round  the  Sun,  at  the  rate  of  75,000  miles 
in  an  hour  :  how  many  miles  does  sh«  travel  in  one  of 
her  years,  or  in  22bi  days  ?  Ans.  410.850.000 

Ex.  5.  The  Earth  travels,  in  her  annual  course,  at  the 
rate  of  68.400  miles  in  an  hour ;  how  many  mUes  there- 
fore do  we  move  in  a  second  ?  Ans.  ly 


S8  MISCELLANEOUS    QUESTIONS. 

Ex.  6.  There  are  in  the  Old  Testamftnt  89  hooks,  and 
929  chapters,  and  in  the  New  there  are  27  books,  and  260 
chapters:  how  many  books  and  chapters  are  there  in  the 
Bible  ?  Ans.  66  books,  and  1 189  chapters. 

Ex.  7.  There  are  23214  verses  in  the  Old  Testament, 
and  7959  in  the  New  i  how  much  therefore  do  the  verses 
in  the  former  exceed  those  in  the  latter  ?   Ans.  15255 

Ex.  8,  There  are  592439  words  in  the  Old  Testament, 
and  181253  in  the  New  ;  how  many  words  are  there  in 
the  Bible  ?  Ans.  773692 

Ex.  9.  In  the  Old  Testament  there  are  2,728,100  let- 
ters,  and  in  the  New  there  are  838.380 :  what  are  the 
sum  and  difference  of  these  two  numbers  ? 

Ans.  3.566.480  sum,   1.889.720  difference. 

Ex.  10.  There  are  in  the  Bible  3.566.480  letters  :  how 
}ong  would  a  person  be  in  counting  tJiem,  supposing  he 
could  count  200  in  a  minute  }   Ans.  297  hrs.  12  minutes. 

Ex.  11.  A  printer  charges  54^.  for  every  J  000  letters 
that  he  sets  up  :  how  many  thousand  must  he  set  up  to 
earn  IL  15s.  per  week  ?  Ans.  80,000 

Ex.  12.  If  a  printer  set  up  8500  per  day,  how  long 
would  he  be  in  composing  the  Old  Testament,  and  how 
long  in  composing  the  whole  Bible  ?     See  Ex.  9  and  10. 
Ans.  321  da^s  Old  Test,  and  419^  Bible,  nearly. 

Ex.  13.  If  a  printer  be  desired  to  set  up  the  Bible  in 
Latin,  how  much  would  he  earn  in  the  business,  at  the 
rate  o^  5ldr  per  lOOO  letters,  supposing  there  are  as  ma- 
ny letters  in  the  Latin  as  there  are  in  the  English  ? 


Ans.  85/.  8s.  Wld. 


Ex.  14.  If  there  be  as  many  letters  in  the  Greek  Tes- 
tament as  there  are  in  the  English,  how  much  would  a 
printer  earn  in  setting  it  up  at  Sid.  per  thousand  ? 

Ans.  SOL  1 1  s.  3ld.  ^o* 

Ex.  15.  The  name  of  Jehovah  occurs  6855  times  in 
the  Old  Testament  :  what  proportion  therefore  does  this 
word  bear  to  all  the  other  words  in  that  book  ? 

Ans.  862  nearly. 

Ex.  16.  The  word  and  occurs  in  the  Bible  46227 
times  :  what  proportion  does  that  bear  to  the  other 
words?  See  Answer  to  Ex.  8.  Ans.  17  nearly. 


MISCELLANEOUS  QUESTIONS.  89 

Ex,  17.  There. are  in  tlie  northern  side  of  London  126 
houses  newly  built,  and  unlet,  the  average  rent  of  which 
is  85^. ;  and  75  houses  at  50L  each,  and  68  at  30  guineas 
each :  what  is  the  total  annual  loss  of  these  empty  houses 
to  the  proprietors  ?  Ans.  16602/. 

Ex.  18.  There  are  1100  hackney  coaches  in  London, 
each  of  which  earns  on  an  average  18s.  per  day  :  how 
much  is  expended  weekly,  daily,  and  annually,  on 
these  vehicles,  Sundays  excepted  ?  Ans.  990^  per  day. 
5940/»  per  week.  308880/.   per  annum. 

Ex.  19.  What  are  256  reams  of  paper  worth,  at  33s. 
6d.  per  ream  ?  Ans.  428/.   16s. 

Ex.  20.  Fifty  thousand  larks  have  been  sold  in  a  sin- 
gle season  in  London :  what  did  they  fetch,  supposing 
they  were  bought  at  lid.  each  ?       Ans.  26oZ.  8s.  4>d. 

Ex.  21.  The  circumference  of  the  Earth,  in  the  lati- 
tude of  London,  is  15,120  miles,  which  is  the  space  we 
pass  over  in  24  hours,  by  the  diurnal  motion  of  the  earth : 
how  much  space  do  we  pass  over  in  a  minute  ? 

Ans,  lOa  miles. 

Ex.  22.  Three  thousand  ounces  of  gold  are  imported 
into  England  annually  :  how  many  pounds  and  grains 
are  imported  in  50  years,  at  this  rate,  and  what  is  the 
value  of  it  at  3/.  18s.  per  ounce  ?  Ans.  12.500  pounds, 
72.000,000  grains,  and  585,000/.  value. 

Ex.  23.  To  work  the  silver  mine^  in  South  America, 
40,000  negroes  are  imported  annually :  how  many  of 
these  poor  creatures  have  perished  in  this  work  during 
the  last  century  ?  Ans.  4.000,000 

Ex.  24.  The  duty  on  hops  amounted,  at  lid.  per  lb. 
in  a  certain  year,  to  26,357/.  9s.  9c/.  :  how  many  hops 
were  grown  that  season  } 

Ans.  1882  tons,  13  cwt.  2  qrs.  6  lb. 

Ex.  25.  The  battering  ram  employed  by  Titus  to  de- 
molish the  walls  of  Jerusalem,  weighed  100,000  lbs. : 
how  many  tons  did  it  contain  ? 

Ans.  44  tons,  12cwt.  3  qrs.  12.  lb. 

Ex.  26.  The  copper  mines  in  the  island  of  Anglesey 
produce  1500  tons  annually,  and  those  in  Cornwall  4000 
tons :  what  is  the  value  of  the  whole  at  9ld.  per  lb.  ? 

Ans.    487,666/.   13s.  4rf. 
8* 


90  MISCELLANEOUS    QUESTIONS. 

Ex.  27.  Mr.  Bolton  coined  40,000,000  penny  pieces, 
each  weighing  an  ounce :  how  many  pounds  of  copper 
were  used  for  them  :  how  much  was  the  value  of  these  in 
pounds  sterling ;  and  what  was  gained  by  this  coinage, 
supposing  the  copper  and  expense  of  coining  to  be  esti- 
mated at  li2ld.  per  pound  ? 

Ans.  2.500.000  lbs.  130,208/.  ds.  Sd.  36,458Z.  6s.  8.d 

Ex.  28.  In  the  year  1794,  43,259,746  yards  of  Irish 
linen  were  exported  from  Ireland  :  how  many  packages 
did  they  make,  each  package  containing  20  pieces,  and 
each  piece  26^  yards  ?  How  many  shirts  would  this  linen 
make,  at  the  rate  of  31  yards  per  shirt  ? 

Ans.  81.622  pack.  86  yds.     11.535.932i*6. 

Ex.  29.  The  circumference  of  the  earth  is  estimated  at 
24,912  miles:  how  many  barley-corns,  (three  of  which 
make  an  inch,)  would  fill  up  this  space  ? 

Ans.  4.735.272.^60 

Ex.  30.  The  territory  of  the  United  States  of  America 
contains  a  million  of  square  miles,  or  640  millions  of 
square  acres  :  of  these,  about  56  millions  are  water  : 
what  number  of  acres,  roods,  and  perches  of  land,  do  the 
United  States  contain,  and  how  many  inhabitants  will 
they  support,  allowing  to  each  4a  acres  ? 

Ans.   129,777.777. 

Ex.  31.  There  are  now  in  England,  Scotland,  and 
AVales,  23  millious  of  acres  of  waste  land  :  how  many 
farms  might  these  be  divided  into,  allowing  to  each  75 
acres  : — and  allowing  5  persons  to  each  farm,  how  many 
souls  would  these  waste  acres  support  ? 

Ans.    306.606  farms  50  acres.   15.333.333  inha. 

Ex.  32.  Between  the  5th  of  July,  1810,  and  the  same 
day,  1811,  there  were  brewed,  by  12  brewers  only,  939, 
900  barrels  of  porter  :  how  much  would  this  quantity  sell 
for  when  retailed  out  at  5d.  per  qt.  allowing  36  gals,  to 
the  barrel  ?  Ans.  2.819.700/. 

Ex.  33.  How  many  hours,  minutes,  and  seconds  have 
elapsed  since  the  birth  of  Christ,  which  is  1808  years, 
supposing  365^  days  in  a  year  ?  Ans.  15.848.928  ho. 
950,935,680  rain.  57,056,140,800  sec. 


MISCELLANEOUS  QUESTIONS.  91 

Ex.  Si.  It  is  said  the  Small-pox  carries  off  in  London, 
by  death,  50  persons  in  a  week  :  how  many  (if  the  dis- 
ease is  not  checked)  will  it  destroy  in  ten  years  ? 

Ans.26.000 
Ex.  35.  There  are  about  10,540  tons  of  cheese  import- 
ed into  London  annually  :   how  much  do  they  sell  for  at 
the  average  price  of  Tgfl^.   per  lb.  .?  Ans.  7.^7.800/. 

Ex.  39.  It  is  computed  that  there  are  50,000  tons  of 
butter  annually  consumed  in  London  :  what  is  the  ex- 
pense, supposing  the  average  price  lOld.  per  lb. '.^ 

Ans.  5.016.656^.  13s.  4-d, 
Ex.  37.  About  120,000  persons  are  employed  in  the 
cotton  trade ;  if  of  these  one-fourth  are  men,  who  earn 
3s.  6d.  a  day,  and  one-fourth  women,  who  earn  Is.  \d, 
a  day,  and  the  rest  children,  who  earn,  each,  3s.  per 
week,  how  much  is  earned  by  manual  l^our  in  the  cot- 
ton manufacture  every  year  ?  Ans.  2,613,000 

Ex.  38.  There  have  been  20,000,000  lbs.  of  tea  im- 
ported in  a  single  year  from  China  ;  what  was  the  value 
of  it,  supposing  the  average  price  4s.  9t/.  per  lb. 

Ans.  4.750,000/. 
Ex.  39.  The  consumption  of  tobacco  in  this  country  is 
about  169,000  cwt. ;  how  much  is  expended  on  this  arti- 
cle at  \ld.  per  oz.  ?  Ans.  1.577,333/.  6s.  8d. 

Ex.  41.  The  consumption  of  milk  is  not  less  than- 
6,980,000  gallons  annually  in  London  ;  how  much  is  ex» 
pended  on  this  article  at  Sets,  per  pint  ? 

Ans.  1,675,200  dollars. 
Ex.  42.  The  iron  rails  round  St   Paul's  cost  11,202/. 
Os.  6c?.,  and   they  weighed  200  tons  and  81  lbs.  ;  what 
was  the  iron  charged  per  lb.  .^  Ans.  6d.  per  lb. 

Ex.  43.  Westminster-bridge  cost  389,500/.  in  build- 
ing ;  how  soon  would  it  have  been  paid  for  by  foot  pas- 
sengers, at  a  halfpenny  each,  supposing  2420  went  over 
each  day  ?  Ans.  21 1  years,  241  ^  days. 


92 


REDUCTION. 


Reduction  is  the  method  of  converting  numbers  from 
one  nauie,  or  denomination,  to  another  of  the  same  va- 
lue; and  it  is  divided  into  Reduction  descending^  and 
Reduction  ascending. 

When  numbers  of  a  higher  denomination  are  to  be 
brought  to  a  lower,  it  is  called  Reduction  descending^ 
and  it  is  pei  formed  bv  Multiplication. 

When  numbers  of  a  lower  denomination  are  to  be 
brought  to  a  higher  denomination,  it  is  called  Reduction 
ascending,  and  is  performed  by  Division. 


REDUCTION  DESCENDING, 

OR  CONVERTrXG  GREAT  INTO    SMALL. 

Rule.  Multiply  the  given  number  by  as  many  of  the 
lower  denomination  as  make  one  of  the  higher. 

Thus,  in  reducing  55l.  into  shillings,  I  multiply  the 
55  by  20,  and  the  answer  is  1 100  shillings  ;  in  both  ca- 
ses the  value  is  the  same,  that  is,  551.  is  equal  to  11 00 
shillinss. 


REDUCTION  ASCENDING, 

OR  CONVERTING  SMALL  INTO  GREAT, 

Rule.  Divide  by  as  many  of  the  lower  denomination 
as  make  one  of  the  next  higher. 

Thus,  in  bringing  890  pence  into  shillings,  I  divide 
the  number  by  12,  and  the  answer  is  74  shillings  and  two 
pence  over. 


heduction,  53 


EXAMPLES. 

L,    s,  d. 

Ex.  1.  Reduce  29     6     8^  into  farthings. 
20 

586  shillings 
12 


7040  pence 

4 

Answer  28 1 Q3  farthings. 

Ix.  2.    In  28163  farthings  how  many  pounds  sterling  ? 

4)28163 


12)7040--| 

2,0)58,6--8c?. 

Ans.L.29  6     8| 
Ex.  3.  Reduce  37  Dimes  to  mills.     Ans.  3700  Mills. 
4.  Reduce  53  dollars  to  cents.     Ans.  5300  cents. 
5*  Reduce  163  eagles  to  dollars.     Ans.  1630  dollars. 

6.  Reduce  74  dollars  to  dimes.    ^Ans.  740  dimes. 

7.  Reduce  217  dollars  to  mills.     Ans.  217000  mills. 

8.  Reduce  35  eaj2;les  to  mills.     Ans.  350000 

9.  Reduce  28  shillings  to  pence.       Ans.  336  Pence. 
JO.  Bring56  pounds  into  shillings.      Ans.  1120  shills, 

11.  Reduce  672  pence  into   farthings.        Ans.   2688 
farthings. 

12.  How  many  pence  are  there  in  105^  ?  Ans.  25200 
pence. 

13.  In  1000  guineas  how  many  shillings  ?  Ans.  210OO 
shillings. 

14.  In  4704^how  many  pence  ?  Ans  1 1 23960  Pence. 

15.  In  3995^.  how  many  farthings  ?  Ans.  3835200  Far. 

16.  In  7968  guineas,  how  many  farthings  ?     Answer, 
8031744  farthings. 

17.  How  many  farthings  are  there  in  75  guineas  } 

Ans.  75600  farthings. 


94  TROY  WEIGHT. 

18.  Reduce  576Z.  into  farthings.      Ans.  552960  far, 

19.  In  99/.  hoM  n)an)'  shillings,  pence,  and  farthings  ? 
Ans.  1980  shilings,  23760  pence,  and  95040  farthings. 

20.  Reduce  567/    &s.    9^.  into  farthings.     Answer. 
544790  farthings. 

21     How   many  halfpence  are  there  in  157/.  7s,  7^. 
An,«.  75543  halfpence. 

22.  fn  1084890  pence,  how  many  pounds  ?     Answer. 
^b2(jt.  Is.  bcL 

23.  in  84^0896  pence,  how  many  guineas  ?     Answer. 
33376  guineas  l2>. 

24^.  in  4808764  farthings,  how  many  pounds  ?    Ans. 
5009/.  2s.  7rf. 

25.  How  many  seven-shilling  pieces  are  there  in  a 
thousand  guineas  ^     Ans.  3000. 

26.  How  many  groats  are  there  in  a  hundred  guineas  ? 
Ans.  6300  groats. 

27.  Bring  3 110456  pence  into  groats.     Ans.  777614 
groats. 

28.  How  many  crown-pieces  are  there   in   79/.   15s.  ? 
Ans.  319  crowns. 

29.  How  many  half-crowns  are  there  in  85/.  \2s\  6d,  ? 
Ans.  685  half-crowns. 

30.  In  769  guineas,  how  many  sixpences  I    Answer. 
32298  sixpences  ? 

TROY, 
OR,  GOLD  SMITHS'  WEIGHT, 

lb.  oz.  dwt.  gr. 
Ex.  1.  Reduce  3    9     6     18  to  grains, 
12 

45 

20 


906 
24 
I 

3632 
1813 

21763 


TROY    WEIGHT. 


95 


Ex.  2.  How  many  pounds  Troy  are  there  in  a  million 

of  graius  ? 
4)1,000000 

6)250000 


2.0;4-U6 .6 — 4  ZI 16  grains. 


12V20^3— 6 

175—7        Answer  173  lbs.  7  oz.  6  dwts.  16  grs. 

Ex.    3.    In  36  lb.    10  oz.    12  dwts.   16  grs.  how  many 
grains?  Ans.  212464  j^rains. 

4.  How  many  pounds  troy  are  there  in  5987  penny- 
weights ?  Ans.  24  lb.  1 1  oz.  7  dwts. 

5.  In  1434  lb.  0  oz.  0  dwts.  19  grs.  how  many  a;rains  ? 

Ans.  8259859  grs. 

6.  How  many  pounds  are  there  in  45065  grains  ? 

Ans.  7  lb.  9  oz.  17  dwts    17  grs. 

7.  Reduce  105  lbs.  troy  into  grains.      Ans.  604800 

8.  In  495  spo(ms,  weighing  103  lbs.  1  oz.   lu  dwts., 
how  many  grains  ?  *        Ans.  594000 


Si 


96  AVOIRDUPOIS   WEIGHT. 

AVOIRDUPOIS, 
OR  GROCERS'  WEIGHT, 

Ex.  1.  How  many  drams  are  there  in  225  tons,  17  cwt. 
3qrs.  241b.  12  oz.  8  dr.  ? 

tons,  cwt,  qr«  lb.  oz.  dr. 
225       17.    3    24.  12     8 
20 

4517 
4 

18071 
28 

144-572 
3r)l44 

506012 
16 

SOS 607 4 
506(1 13 


8096204 
16 


48577232 

8096204 

Answer,  129539272  drams.  • 

129539272 


AVOllRSUPOIS    WEI6H9*. 


S7 


Ex.  3.  How  many  tons  are  there  in  259078544  drams  ?• 
4)259078544 


v4>64769636 


4)16192409 


4)4048102  —  1 


7)1012025 


—  1  7      oz. 


4)144575 

lb. 

4)36143  —  3=21 

2.0)903.5—3 


451    15  3  21   ^ 

Ans.  451  tons,  15  cwt.  3  qrs.  21  lb.  9  oz. 

Ex.  3.  In  179  cwt,  how  many  pounds  ?  Ans.  20048  lb. 

4.  Reduce  8345  tons  into  quarters.  Ans.  607600  qrs. 

5.  How  many  ounces  are  there  in  4  tnns,  15  cwt.  2qrg. 
12lb.  ?  Ans.  171328  ounces. 

6.  In  233076  ounces  of  sugar,  how  many  cwt.  ? 

Ans.  130  cwt.  Oqr.  71b.  4  oz. 

7.  How  many  drams  are  there  in  53  tons,  14  cwt.  1  qr. 
4^ lb.  14 oz.  8  dr.?  Ans.  3()804200  drams. 

8.  In  3-2384818  drams,  how  many  tons  weight  ? 

Ans.  56  tons,  9  cwt,  1  qu  27  lb,  3  ©z.  2  dr* 


98  apothecaries'  weight. 


APOTHECARIES*  WEIGHT. 

Ex.  I.  How  many  grains  are  there  in  2  lb.  5  oz.  4  dr* 
1  scr.  17  gr.  ? 

lb.  oz.  dr.  scr.  gr. 
t    5     4      1     17 

29 
8 


236 
3 

709 
20 

14i97        Answer  14197  grains. 
£^»  2.  In  42591  grains,  how  many  pounds  \^ 
2.0)4259.1 


3)2129  —  11 

8)7(:9— .2 
22)88  —  5 


7     4     5     2     11 
Answer  -  71b.  4  oz.  5  dr.  2  scr.  11  gv. 
Ex.  3.  In  51  lb.  2  oz.  of  rhubarb,  how  many  scruplesi 

Aiis.   14736  scrup. 
4.  Id  234876  grains,  how  many  pounds  ? 

Ans.  40  lb.  9  oz.  2  dr.  1  scr..l6gr. 
,5.  How  many  pounds  are  there  in  1000  6^.  of -opium  ? 

Ans.  83  lb.  4  <  z. 

6.  In  239  lb.  9  oz.  2  dr.  2  scr.  14  gr.,  how  many  gis.  .>^ 

Ans.    lS8n64grs. 

7.  How  nany  scruples  are  there  in  or.e  hundred  and 
three  ounces  of  Peruvian  bark  ?  Ans.  2472  scrup. 

8.  In  J20794  grains,  how  many  pounds? 

Ans.  22  lb.  0  oz.  1  scr.  0  dr.  14  gr. 


LONG    MEASURE, 


LONG  MEASURE. 

Ex.  1.  How  many  yards  are  there  between  London 
and  Bath,  the  distance  of  which  is  JOS  mile&? 
JOS 
8 

864 
40 


345^0 
5| 

172800 
17280 


190080    Answer  190080  yds, 

Ex.  2.  In  f  60S29  feet,  how  many  leagues  ? 
3)760329 


253443 
2 


11)506886 


4.0)4608.0  — .  6  IZ  3 
8)1151^  —  0 
3)144  —  0 


43 
Ans.  48  lea.  0  m.  0  fur.  0  p;  3  yards. 
Ex.  3.  How  many  inches  are  there  in  1009  miles  ^ 
^  Ans.  63930240  inches. 

4.  Reduce  57  ra.  4  fur.  38  p.  3  yds.  2  ft.  3  in.  1  b.c- 
into  barley -curns  ?  Ans.   10952378  b.  corns. 

5.  In  100004  poles,  how  many  inches  r 

Ans.    19800792  inches. 


180  JELOTH    MEASURER 

Ex.  6.  In  40y683  feet,  how  many  furlongs  ? 

Ans.  620  fur.  161  yards. 

7.  How  often  will  the  wheel  of  a  coach  turn  round  in 
going  from  London  to  Sheffield,  or  in  160  miles,  suppos- 
ing the  circiunference  of  the  wheel  to  be  J  6  feet } 

Ans.   32800  times. 

8.  Suppose  on  an  average  I  step  two  feet  and  a  half; 
how  many  steps  shall  I  take  in  walking  from  London  to 
Richmond,  a  distance  of  10  miles  ?     Ans.  21 120  steps. 


CLOTH  MEASURE. 
Ex.  I.  How  many  inches  in  length  are  there  in  15^ 
^8  English  of  cambrick  ? 

15& 

780 
4 

SI  20 


7020      Answer  7026  inches; 
Ex.  2.  In  1000  inches  of  cotton,  how  many  yards  arc 
there  ? 


27  3  0  1 
Ans.  27  yds.  3  qr.  On.  1  in; 
Ex.  3.  How  many  English  ells  are  therein  three  thou- 
sand and  fifty -five  nails  ?      Ans.  152  E.e.  3  qrs.  3  n. 
4.  In  15  yds.  2  qr*  3  n.  1  in.,  how  many  half  inches  ? 

Ans.   1131^  half  inches. 
6,  How  many  inches  are  there  in  10056  yards  ? 

Ans.  362016  inches. 
6.  Reduce  546  English  ells  to  nails.  Ans.  109:^0  n\sh 


SQUARE)   OR   LAND    MEASURE.  101 


SQUARE,  OR  LAND  MEASURE. 

Ex.  1.  How  many  yards  are  there  in  5604  acres  ? 
5604 
4 

224- Id 

40 


896640 
50| 

26899200 
224160 


27123360      Ans.  27123360  yds, 

Ex.  2.  In  6534  square  feet,  how  many  perches  ? 
9)6.534 


50^ 
4 

7xi6 
4 

121 

121)2004(24 
242 

484 
484 

Answer  24  perches. 

Ex.  3.  How  many  roods  are  there  in  382  perches  ? 

Ans.  9  roods.  22  perches* 

4.  In  561  acres  of  ground  how  iiarvy  (lerch.  and  vds.  ? 

Ans.  89760  ot-r.  271 5240  yds. 

5.  In  2967 4f)0  inches  how  m  iiy  acres  ? 

Ans.  Not  quite  a  an  acre,  l)i*.*ng  ordy  2289  yds.  5*^. 

6.  How  many  perches  are  there  in  997  acr.  3  rd.  I0j>.? 

Ans.  lo9610  perches. 

9* 


102  OUBIC,   OR    SOLID    MEASURE. 

CUBIC,  OR  SOLID  MEASURE:. 

£^.  1.  In  36  solid  ^ards,  how  many  inches  E 
36 
27 

72 

972 
1728 

7775 
1944 
6804 
972 


Answer    -     1 6796 1 6  inches. 

'3.  In  1259712  solid  inches,  how  many  yards  ? 

Ans.  27  yardat 


LIQUID  MEASURE. 


•Ex. 

J.  How  many  gallons  j 
o 
2 

10 

63 

Answer    .     630 

ire  there  in 
gallons. 

5  pipes  of' 

wine? 

Ex.  2.  In  700C.  pints, 

2;70Q6 

how  many 

gallons  ? 

4)3503 

876  3      Ans.  875  gal.  S  qts. 


DRY   MEASURE. 


103 


Ex.  3.  In  31490  pnts,  how  many  gallons  ? 

Ans.  3936  gal.  1  qt. 

4.  In  3  tuns,   I  hhd.  49  gallons  of  claret,   how  many 
quarts  ?  Ans.  347^1  quarts 

5.  Mow  many  tuns  of  port  wine  are  therein  46088 
gallons.^  Ans.  183  tuns,  3  hhd. 


65  gal. 


miv  MEASURE. 


Ba.  1.  In  79  pks.  how  many  pts.  ?   Ans.  12^64  pts. 

2.  How  many  bushels  are  there  in  7649  pints  ? 

Ans.   119  bush.  2  pks.  1  pt. 

3.  How  many  pts.  are  there  in  23  bush.  3  pks.  2  qts.? 

Ans.  1324  pts. 

4.  In  3  pks.  and  1  gal    how  many  qts.  ?  Ans.  28  qts. 

5.  How  many  pks.  are  there  in  187406  quarts  ? 

Ans.  23425  pks.  1  gal.  2  qts. 


COMMERCIAL  NUxMBERS, 

OR    ARTICLES    SOLD    BY    TALE. 


1 2  articles  of  any  kind    - 

13  ditto 

12  dozen        -         -        - 
20  articles  of  any  kind    - 

5  score 

6  score 
12  score 

6  dozen  skins  of  parchnnint    - 
72  words  in  Common  l/ 
SO  _  in  the  Kxcheq* 

9o i.i  Chancery 

34  -ihe"t>  of  paper 

520  quires 

21a  quires,  or  3l6  sheets 

2  reams 


hmen 
3m: 


1  dozen 

1  lon^  dozen 

1  gross 

I  score 

1  hundred 

1  great  hundred 

1  pack  of  wool 

1  roll 

I  sheet 

1  ditto 

1  ditto 

1  fiuire 

1  ream 

1   do.  printer's 

1  bundle 


104  ARTICLES   BT  TAtE. 

Folio  is  the  largest  size  of  books,  of  which, 

2  leaves,  or  4  pages,  make  a  sheet. 
Quarto,  4io.      -       4  leaves,  or  8  pases,  make  a  sheet^ 
Octavo,  8vo.      -       8  h-aves,  or  16  pages,  ditto. 
Duodecimo,12no.  12  leaves,  or  24  pages,  ditto. 
Octodecimo,! Brno.  18  leaves,  or  36  pages,  ditto* 


EXAMPLES. 

Ex.  1.  How  many  long  dozeu  ^^..^  there  in  ten  thou- 
sand oranges  ?  Ans.  769  doz»  ^  ^^anq-es. 

Ex.  2.  How  many  gross  are  there  in  one  hundred  ana 
fifty  thousand  corks  ?     Ans.   1041  gross,  8  doz.  corks. 

Ex.  3.  In  seventy  thousmd  quills,  how  many  great 
lundref's  are  there  ?        Ans.  583  hundreds  40  quills. 

Kx.  4.  I  have  a  deed  containing  4  skins  of  parchirtent^ 
and  each  skin  contains*  850  words  ;  for  how  many  sheets 
shall  I  have  to  pay  the  person  who  copies  it,  reckoning 
according  to  the  common  law  charge  ? 

Ans.  47  sheets  and  16  words. 

Ex.  5.  The  writing  of  an  Kxchequer  cause  occupies 
315  sheets  ;  for  how  many,  words  shall  I  have  to  pay  the 
clerk  who  copies  it  for  me  ?  Ans.  25200  words. 

Ex.  6.  A  suit  has  been  four  years  in  chancery,  and  I 
wish  to  have  a  copy  of  all  the  proceedings  ;  for  how  ma- 
ny sheets  shall  I  pay.  supposing  it  occupies  1264  skins  of 
parchment,  and  each  skin  6Q0  words  ? 

Ans.  9690  sheets  and  60  words. 

Ex.  7.  How  many  sheets  are  there  in  4o  reams  of 
paper?  Ans.  19200  sheets. 

Ex  8.  How  many  common  reams  of  paper  are  there 
in  ten  thousand  printer's  reams  ?     Ans.    t0750  reams. 

tCx.  9.  W  hat  number  of  sheets  less  are  tliere  in  500 
common  rea?MS  of  paper,  thaik  there  are  in  the  same  num- 
ber of  printer's  reaios  ?  Ans.   18000  sheets. 

Ex.  10.  What  number  of  pages  are  there  in  a  folio 
containing  2!l  sheets  ?  Ans.  844  pages.. 

Kx  II.  What  will  be  the  difference  in  the  numher  Of 
pages,  whether  1  print  in  l2mo.  or  iSno.,  supposing  my 
work^wiil  make  lourteeo  sheets  ?        Aus.  ids  pages. 


TIME.  1Q5 

Ex.  12.  What  numbers  of  words  are  there  in  Dr.  Gre- 
gory's Dictionary  of  Arts  an\l  Sciences,  which  contains 
240  sheets  4to.,  and  each  page  contains  1848  words  ? 

Ans.  3548160  words. 

Ex.  13  How  many  reams  of  paper  ivere  used  in  print- 
in  jj  that  Dictionary,  six  thousand  copies  having  been 
taken  off?  Ans.  3000  reams. 

Kx.  14.  How  many  pens  were  used  in  writing  the 
said  Dictionary,  supposing  each  pen  to  write  840  words  ? 

Ans.  4224  pens. 


TIME. 

Ex.  1.  In  4199  days,  how  many  months  of  28  days 
each,  and  years  of  365  days  each  ? 

Ans."  149  ms.  27  days;  or  ii  yrs.  and  i  nearly. 

2.  Ucduco  ISO  days  to  hours  and  minutes  ? 

Ans.  3600  hours,  216000  minutes. 

3.  In  70  years  how  many  days,  supposii>g  each  year 
to  consist  of  365^  da^s  ?  Ans.  25567  days  and  a. 

4.  How  many  minutes,  hours,  and  days  are  there  ia 
5960034  seconds  ? 

Ans.  99333  min.  1655  ho.  or  68d.  23ho.  33min.  54s. 

5.  How  many  minutes  are  there  in  1808  years,  allow- 
ing 365^  <iays  make  one  year  ?      Ans.  950935680  min. 

6.  How  many  seconds  has  a  boy  lived,  who  is  12  years, 
9  months,  and  13  days  old,  reckoning  13  lunar  months 
of  28. days  each  to  a  year  ?  Ans.  400291200  sec. 

ASTRONOMY. 

TABLE. 

60  second  (60")     -        -  -  1  minute,  V 

6o  minutes     -        -        -  -  l  degree,    1° 

30  dejirees     -        -       -  -  1  sign 

12  signs,  or  360  degrees  -  1  great  circle. 

Ex.  I,  In  185  degrees  how  many  minutes  and  seconds  r 

Ans.   11100  min.  666000  seconds. 
2.   How  many  degrees  are  there  in  five  thousand  and 
fifty-five  seconds  ?  Ans.  I  deg.  24  luin.  15  sec. 


106  MISCELLANEOUS    EXAMPLES. 

3.  How  many  seconds  are  there  in  a  great  circle  ? 

Ans.  J 296009  seconds. 

4.  In  548056  seconds,  how  many  signs  ? 

Ans.  5  signs  2  degrees  14  minutes  16  seconds. 

5.  How  many  seconds  are  there  in  9  sig.  4  deg.  56 
min.  and  56  sec  ?  Ans.  989756  seconds. 

6.  In  700809  seconds,  how  many  degrees  ? 

Ans.  194  deg.  40  min.  9  sec. 


MISCELLANEOUS    EXAMPLES. 

Ex.  I.  In  195  pounds,  how  many  shillings,  pence  and 
farthings  .P     Ans.   3900  shil.  46800  pence  187200  far. 

Ex.  2.  In  77  guineas,  how  many  shillings,  pence  and 
farthings?       Ane.  ifil7  shil.  19404  pence,  77616  far. 

Ex.  3.  How  many  crowns,  half-crowns,  shillings  and 
sixpences,  are  there  in  354  pounds  ? 

Ans.   141j5  crowns,  2832  half-c.  7080  shil.  I4l60  sixp. 

Ex.  4.  In  4432127  farthings,  how  many  pence,  shiU 
lings,  pounds,  and  guineas  ?       Ans.   1108031  pence  |. 
92335s.   lid.     4616/.   I5s.  4396  guineas  I9s. 

Ex.  5.  In  14  ingots  of  silver,  each  weighing  27  oz. 
5  dwts.,  how  many  grains  ?  Ans.  183120  grains. 

Ex.  6.  In  three  dozen  of  table  spoons,  each  weighing 
{S  oz.  and  9  dwts.,  how  many  pounds  > 

Ans.  r  lb.  4  oz.  4  dwts. 

Ex.  7.  In  78  bags  of  hops,  each  weighing  3  cwt.  how 
many  pounds  ?  Ans.  26208  lb. 

Ex.  8.  How  many  pounds  and  cwts.  of  tobacco  are 
there  in  73  hogsheads,  each  containing  3  cwt.  1  qr.  14  lb.? 
Ans.   28350  lb.  253  cwt.  0  qr.   14  lb. 

Ex.  9.  In  98465  inches  of  broad  cloth,  how  many  yds. 
and  ells  ?  ^Ans.  2735  yds.  5  in.  ;  2188  ells,  5  in. 

Ex.  10.  In  five  thousana  yards  of  cloth,  how  many 
nails  ?  Ans.  80000  nails. 

Ex.  11.  How  many  inches  are  there  between  London 
and  Bristol,  a  distance  of  120  miles  ?  Ans.  7603200  in. 

Ex.  12.  How  many  barley-corns  will  reach  round  the 
earth,  which  is  a  great  circle  of  360  detjrees,  and  each 
degree  contains  69a  miles.     And  how  many  quarters  Cff 


MISCELLANEOUS   S1CAMPLES.  107 

barley  would  be  necessary  to  perform  this,  supposing 
9200  barley-corns  to  fill  a  pint  measure  ? 

Ans.  4"53^Ol600  b.c;  1009  qrs.  5  bush.  6  pts.  8800  b.c. 

Ex.  13.  How  often  will  a  wheel  turn  ingoing  trom 
London  to  York,  a  <iistance  of  198  miles,  if  the  wheel 
be  21  yards  in  circumference  ?        Ans.  139392  times. 

Ex.  14.  How  many  perches  are  there  in  a  field  con- 
taining 105  acres  .'^  Ans.    1(5800  perches. 

Ex.  15.  If  a  field  of  5  acres  be  taken.  fi*'>ra  one  of  56 
acres,  how  many  square  yards  will  remain  ? 

Ans.  246840  sq.  yards. 

Ex.  16.  How  many  pints  anxi  gallons*  are  there  in  39 
hogsheads  of  cyder  ?  Ans.    19666  pts. ;  2457  gal. 

Ex.  17.  How  many  minutes  have  elapsed  suice  the 
creation  of  the  world  to  the  present  time,  1808  inclusive, 
supposing  the  world  to  have  been  created  1004  years  be- 
fore the  birth  of  Christ,  and  each  year  to  consist  of 
SQ5l  days  ?  Ans.  3056879520  minutes, 


AMERICAN  COIN. 

TABLE. 

Currency  s,  d.  Fed.money, 

In  Maryland    Pennsylvania,  >  ,  g  make  1  DoL 
Delaware  and  Mew  Jersey,    y 

ISew  England  and  Virginia.  6  0        ditto. 

New  York  and  North  Carolina.  8  0        ditto. 

South  Carolina  and  Georgia.  4  8        ditto. 

Canada  and  Nova  tScotia.        ^^  5  0'      ditto. 

1.  To  reduce  Maryland,  Pennsylvania,  Delaware  and 
New  Jersey  currencies  to  Fe<ieial  money,  the  value  of 
the  dollar  being  7s.   6d.  or  90  pence. 

Rule.  Reduce  the  given  sum  to  pence  and  divide  by 
90,  the  result  will  be  dollars,  ^npex  a  cypher  and  con- 
tinue the  division  for  cents. 


108 


REDUCTIOlf, 
EXAMPLES. 


Ex.  t.  Reduce  76/.  14s.  6d.  Maryland  currency  to 
edefal  money.  ** 


Federal  money. 

L.  s,  d, 
76.  14.*  6. 
•  20 


9,0)184140 


Ans.  S  204.60 

Ex.  2.  Reduce  237/.  17s.  4rf.  Pennsylvania  currency 
to  Federal  money.  Ans.  634  dols.  SJ  cts. 

3.  Reduce  5217/.  6s  7d.  Delaware  currency  to  Fede- 
ral money  ?  Ans.   lS9l2dols.  87  cts. 

4.  Reduce  673/.  Is.  Qd.  New  Jersey  currency  to  Fede- 
ral money  >  Ans.  1794  dols.  82  cts. 

5.  Redfuce  71.  6s.  8d.  New  Jersey  currency  to  Fede- 
ral money  ?  Ans    l9  dols.  55  cts. 

6.  Reduce  39/.  7s  6d,  Maryland  currency  to  Fede* 
ral  money  ?  Ans.  105  dollars. 

7.  Reduce  48/.  9s.  5d.  Pennsylvania  currency  to  Fe- 
deral money  ?  Ans.  129  dols.  25  cts. 

a 

2.  To  change  Federal  money  to  Maryland,  Pennsyl- 
vania, Uelauare  and  New  Jersey  currencies. 

KvLE.  It  the  given  sum  be  doijars  only,  multiply  by 
90  and  the  result  wilkibe  pence,  but  if  there  should  be 
cents  in  the  given  sum,  multiply  by  90  and  cut  oft  two 
fij^ures  on  the  right  hand,  the  result  will  be  in  pence  also^ 
which  reduce  to  shillings  and  pounds. 

KOTE, 

IF  there  should  be  half-pt-ticc  or  farthings  jn  the  given  sUrt, 
retiiice  it  to  the  lowest  denomination  mentioned,  and  reduce  also 
the  number  ut  pi  nee  in  ont  dollar  to  the  same  denomination,  and 
divide  by  iiii^  tor  the  answer. 


REDUCTION.  109 


EXAMPLES. 

Ex.  How  much  Maryland  currency  in  76  dols.  50  cts.? 
duls,  cts. 
76     50 
90 


12)6885,00 
20)57,3  9 
Answer  -  -  -  2Sl.  I3s.  9d, 

2.  Change  744  dols.  into  Pennsylvania  currency  ? 

Answer  279^ 

3.  In  365  dols.  25cts.  how  much  New-Jersey  cur- 
rency  ?  Ans.   136/.  19*-.  4*c/. 

4.  In  7493  dollars  50  cents,  how  much  Delaware  cur- 
rency ?  Ans.  2810/.   Is.  3d. 

5.  In  627  dollars  75  cents,  how  much   Pennsylvania 
currency  ?  Ans.  235/.  8s.  1^. 

6.  In  134  dollars  60  cents,  how  much  Maryland  cur- 
rency? Ans.  50/.  9s.  6d. 

7.  in  1216  dollars  80  cents,  how  much  Pennsylvania 
currency  ?  Ans.  456/.  6s. 


3.  To  change  New  England  and  Virg;inia  currencies 
to  Federal  iin)ney,  the  value  of  the  dollar  being  6  snil- 
lings  or  72  pence. 

Rule.— If  there  be  pounds  and  shillings  only;  re- 
duce the  given  sum  to  shillings,  and  divide  by  6  :  but  if 
there  be  peiic  also,  retfuce  the  given  sum  to  pence ; 
then  divide  bv  72,  and  the  quotient  will  be  dollars,  an- 
nvx  two  cyphers  to  the  dividend^  and  cuntiuue  the 
operatLun  for  ceuts. 

10 


110  REDUCTION. 


EXAMPLES. 


1.  In  74?.  6s.  8d.  New  England  currency,  how  much 
Federal  money  ? 

L.    s.     d, 
74    6      8 
20 

U86 
12 

dols.  cts. 


72)17840.00(247     77  Answer. 
144 

344  or  thus, 

«IOO0 


288  f8)l7! 

n]     — 

560  (_   9)2 


3U00 


8  247,: 7  cts. 


504 

560 

504 

560 

504 

56 

2.  Tn  64i.  15s.  Virginia  currency  how  much  Federal 
money?  Ans.  215  dollars  «3  cents. 

3.  Tn  327?.    l6s.  4rf.  Virginia  currency,    how   much 
Federal  money  ?  Ans.  1092  dollars  72  cents, 

4.  In   463/.  12s.    9d.   Virginia  currency,  how  much 
Federal  money  ?  Ans.    1545  dollars  45  cents, 

5.  In  579?.   18s.  2d,    New  England   currencv,    how 
murh  Federal  money  ?         Ans.   1933  dollars  2  cents. 

6.  In  6214?.  12s.. 9d.  Virginia  currency,  how    much 
Federal  iN  oney  ?  Ans.  2''7 15  dollars  45  cents. 

7.  In    7  >09?.  13s.  7d.  Virgitiia  currency,  how    much 
Federal  money  ?  Ans.  24365  dollars  59  cents. 


REDUCTION.  H) 

4.  To  chann;e  Federal  money  to  New  England  and 
Vir2;inia  currencres. 

Rule. — Multiply  the  given  number  of  dollars  by  6, 
and  divide  by  20  for  pounds,  or  if  ther*^  be  cents  in 
the  question,  multiply  the  number  of  cents  by  72,  and 
divide  by  100,  the  quotient  will  be  pence,  which  reduce 
to  shillings  and  pounds. 

EXAMPLES. 

Ex.  I.  Change  273  dollars  23  cents  to  New  England 
currency  ? 

27325  cts. 
72 


54630 
loo)      191275 


12)19674,00 
2,0)1(53-9  d 


Answer  -  -  -  8H.  19s.  6i. 

2.  Change  49G  dollars  to  New  England  currency  ? 

Ans.    148/.   rSjf. 

3.  Change  TO  dollars  5o  cents  to  Virginia  currency  ?^ 

Ans.  231.   17  s. 

4.  Change  673  dollars  60  cents  to  Virginia  currency  ? 

Ans.  202/.   Is.  7(1. 

5.  Change  762  dollars  15  cents  to  Virginia  currency  } 

Ans.  228/.   12^.   lOi. 

6.  Change  847  dollars  75  cents  to  Virginia  currency  ? 

Ans.   254/.  OS.  6d. 

7.  Change  1740  dols.  30  cents  to  Virginia  currency  ? 

Ans.  323/.  17 s.  9d. 

5.  To  change  New  York  and  North  Carolina  currencies 
to  Federal  money,  the  value  of  the  dollar  being  8  shil- 
lings or  96  pence. 

KvLE — If  tiie  lowest  denomination  mentioned  in  the 
given  sum  be  shillings,  reduce  it  to  that  denomination 


^12  REDUCTION. 


and  divide  bj  8  ;  but  if  the  lowest  denomination  be 
pence,  reduce  the  given  sum  to  pence  and  divide  by  96, 
the  quotient  will  be  dollars  ;  bring  down  two  cyphers 
and  continue  the  operation  for  cents. 


EXAMPLES. 

Ex.  1.  In  74/.   16s.  New  York  currencj  how  much 
Federal  money  ? 

L,    s. 
74.    16 
20 

8)1496    • 


187  dollars.  Ans. 

2.  In  29/.  17s.  New- York  currency  how  much  Federal 
money  ?     Ans.  74  dolls.  62^  cts. 

3.  In   365/.  7s.  4rf.   New- York  currency  how  much 
Federal  money  ?    Ans.  913  dols.  41  cts. 

4.  In  497/.   16s.  10^.   North  Carolina   currency  how 
much  Federal  money  ?    Ans.  1244  dols.  60  cts. 

5.  In  563/.   12s.  6d,  New-York  currency  how   much 
Federal  money  ?     Ans.  1409  dols.  6  cts, 

6.  In  728/.  ]3s»  9rf.  New-York  currency  how  much 
Federal  money  ?     Ans,  1821  dols.  71  cts. 

7.  In  3674/.  8s.  7d.  North  Carolina  currency  how  much 
Federal  money  ?     Ans.  9186  dols.  7  cts. 


'  «.  To  change  Federal  money  to  New-York  and  North 
Carolina  currencies. 

Rule— Multiply  the  given  number  of  cents  by  96.  and 
divide  the  product  by  100,  the  quotient  will  be  pence, 
which  reduce  to  shillings  and  pounds,  or  if  there  be  dol- 
lars only  in  the  question,  multiply  thorn  by  8  and. divide 
by  20  for  pounds,  the  remainder,  if  any,  will  be  shillings. \ 


REDUCTION.  113 

EXAMPLES. 

Ex.  1.  Reduce  49  dols.  50  cts.  to  New  York  currency. 

cts. 

4950 

96 


207  OO 
1,00)    44550 


J2)475?.00 
2,0)39.6 


Ans.  L19   16 

2.  Reduce  246  dols.  to  North  Carolina  currency. 

Ans.  987.8s 

3.  Reduce  418  dols.  75  cts.  to  New-York  currency. 

Ans.   167/.  10s. 

4.  Reduce  672  dols.  25  cts.  to  North  Carolina  curren- 
cy. Ans.  263^.  ISs. 

5.  Reduce  847  dols.  60  cts.  to  North  Carolina  curren- 
cy. Ans.  339/.  Os.  [)d, 

6.  Reduce  U84  dols.  40  cts.  to  North  Carolina  curren- 
cy. -Ans.  473/     15s\  2/. 

7.  Reduce  2756  dols.  50  cts.  to  New -York  curr<nry. 

Ans.  1102/.  12*-. 


"^  7.  To  change  South  Carolina  and  Georgia  currencies 
to  Federal  money,  the  value  of  the  dollar  being  4.s.  8i  or 
56  pence. 

Rule. — Reduce  the  yjiven  sum  to  pence,  then  divide 
by  56  and  the  quotient  will  l)e  dollars,  bring  down  two 
cyphers  and  continue  the  operation  for  cents. 


10* 


Ill-  REDUCTION. 


EXAMPLES, 


Ex.  1.  In  691,  15  V.  6i.  South  Carolina  currency^  how 
much  Federal  money  ? 


L.     s.    d. 

69    15    Q 
20 


56)16746(299  03 
112 

554 

504 

506 
504 

200 
168 

32 

2.  Tn  864^.  17s.  2c?.  South  Carolina  currency,  how 
much  Federal  money  ?  Ans.  3706  dols.  53  cts. 

C->,  In  92"'/.  I6s.  9</.  Georgia  currency,  how  mucli  Fe- 
deral money  ?  Ans.  39r6  dols.  44  cts. 

4.  In  673/.  I2s  8c?.  Georgia  currency,  how  much  Fe- 
deial  money  ?  Ans.  28«7  dois. 

5.  In  763/.  18s.  irf.  Georgia  currencv,  how  much  Fe- 
deral money  }  Ans.  3273  doh.  87  cts. 

6.  Ill  111/.  I  Is.  lie?.  Georgia  curreiic,  how  much  Fe- 
de«al  money  >  Ans.  478  dols.  26  cts. 

7.  In  5106/.  17s.  4c?»  Georgia  currencv,  how  much  Fe- 
deral money  ?  Ans.  13315  dols.  14  cts. 


V  8.  To  change  Federal  money  to   South  Carolina  and 
Georgia  currencies. 


i^«bi^. 


HEDUOTION.  115 

Rule. — Multinly  the  gjiven  nimber  of  cents  hv  56, 
and  divide  by  100.  the  quotient  will  be  pence,  which  re- 
duce to  shillings  and  pounds,  t. 

EXAMPLES, 

Ex.  I.  How  much  Georgia  currency,  in  216  dols.  SOcts.  ? 
cts. 
21650 
56 


12P900 
1,00)    1  OK  250 


12)12124.,00 
20)101,0  4 


Ans.   L.50   10  4 

2.  How  much   South  Carolina  currency  in   467  dols. 
25  rts.  ?  Ans.  109/.  Os.  6d, 

3.  How  much   South  Carolina  currency  in  762  dols. 
30  cts.  .?  Ans.  177/.  1 7s   4^^. 

4.  How  much  Georgia  currency  in  939  dols.  70  cts.  ? 

Ans.  !^19/.  5>.  3d. 

5.  How  much  Georgia  currency  in  1000  dols.  P 

Ans.  233'  6^,  8. 

6.  How  much  Georgia  currency  in  2 172  dols.  .50  cts.  P 

Anv   506/   18>.  4rf. 

7.  How  much  Georgia  currency  in  9999  dols  99  cts.  ? 

Ans.  2333/.  6^.  id. 


''  9.  To  change  Canada  and  Nova  Scotia  currencies  to 
F'deral  money,  the  value  of  the  dollar  being  5  shillings 
or  60  pence. 

Rule — Reduce'  the  given  sum  to  pence  and  divide   by 
60,  the  quotient  will  be  dolUrs ;  or  if  the  lowest  denouii- 


116  REDUCTION. 

nation  be  shillings,  reduce  to  shillinj^s  and  divide  by  5, 
the  quotient  will  be  dollars  ;  annex  two  cyphers  and  con- 
tinue the  operation  for  cents. 


EXAMPLES. 

Ex.  1.  Reduce  87/.  16.  4rd.  Canada  currency  to  Fede- 
ral money  ^ 


L.     s.     d, 
87     16    * 
20 


«,0)2 107,60 
Ans.  g33I   26  cts. 

2.  Reduce  24  14  to  Federal  money  h 
20 


6)494.00 

Ans.  898  80  cts. 

3.  Reduce  827/.  15s.  Nova  Scotia  currency,  to  Federal 
jnonpy  ?  Ans    S3]  I   dols. 

4.  Ueduce  268/.  I2s.  3d.  Canada  currency,  to  Federal 
money  ?  Ans.  1074  dols  45  cts. 

5.  Reduce  719/.  9s.  2d,  Canada  curr»'ncy,  to  Federal 
money  ?  Ans.  2877  dols.  83  cts. 

6  Reduce  672/.  iOs.  lOrf.  Canada  currency,  to  Fede- 
ral tvoney  f  Ans.  2ti90  dols  16  cts. 

7  Reduce  926/   1  Is.  I  \d,  Canada  currency,  to  Fede- 
ral nu>ney  ?  Ans   3706  dols.  38  cts.     . 

8  Reduce  5119/.  VOs.  id,  Canada  curr'  ncv,io  Federal 
money  ?  Ans.  20479  dols.  91  cts. 


V'^.' 


REDUCTION. 


117 


10.  To  change  Federal  money  to  Canada  and  Nova 
Scotia  currencies. 

Role— Multiply  the  given  number  of  cents  by  60, 
and  divide  the  product  by  100,  the  quotient  wdl  be  pence, 
which  reduce  to  shillingis  and  pounds;  or  it  there  be  dol- 
lars only  in  the  question  multiply  them  by  5,  divide  by 
20  for  pounds,  and  the  remainder  will  be  shillings. 

EXAMPLES. 

Ex.  1.  In  68  dels.  50  cts.  how  much  Nova  Scotia  cur- 
rency ? 

cts. 

6850 

(jO 

l^oo) 

12)4110,00 


2,0)34,2  6 

Ans.  L.17  2  6 

2.  In  124  dols.  25  cts.  how  much  Canada  currency  ? 

Ans.  31/.   Is.  3d. 

3.  In  7648  dols.  how  much  Canada  currency  ? 

Ans.  1912Z. 

4.  In  867  dols.  35  cts.  how  imich  Canada  currency  ? 

Ans  216/.  I6s.9d, 

5.  In  1714  dols,  75  cts.  how   much   Nova  Scotia  cur- 
rency ?  Ans.  428/.   13s.   9d. 

6.  In  6179  dols.  20  cts.  how  much  Canada  currency? 

Ans.  1541./.  16s. 

7.  In  4444  dols.  44  cts.  how  much  Canada  currency  ? 

Ans.  llll/.2i.  2rf. 


(118) 
PROPORTION, 

OR 

THE  RULE  OF  THREE, 


This  rule  is  called  the  Rule  of  Three,  because  by  three 
numbers  being givcMi  we  find  a  fourthj  and  it  is  either  the 
Rule  of  ilirce  Direct  or  Inverse. 


THE  RULE  OF  THREE  DIRECT 

teaches,  from  three  given  numbers  to  find  a  fourth,  which 
slidll  have  the  same  proportion  to  the  second,  as  thetliird 
has  to  the  first  ;  that  is,  if  the  first  be  greater  than  the 
third,  the  second  will  lie  greater  tlian  ihc  fourth  ;  and, 
if  the  first  be  less  than  the  third,  the  second  will  be  less 
than  the  fourth. 

Rule  I.  State  the  question:  that  is,  place  the 
given  numbers  so  that  the  first  and  third  may  be  of  the 
same  kind,  and  the  second  the  same  as  the  number  re- 
quired. 

2.  Bring  the  first  and  third  numbers  into  the  same  de- 
no;nination,  and  the  second  into  the  lowest  denouiina- 
tion  mentioned. 

3.  Multiply  the  second  and  third  numbers  together, 
and  divide  the  product  by  the  first,  and  the  quotient  will 
be  the  answer,  in  the  same  denomination  as  that  in  w  bich 
the  second  number  was  left. 


nULE    OF    THREE    DIRECT.  Il9 

Ex.  1.  What  is  the  value  of  a  pipe  of  wine,  if  5  gal- 
lons cost  4/.    178. 

gal.  L.  s,  pipe. 

5  :  4   17::  1 

20  2 

97  2 

63  « 

126 

97 

882 
1134 

5)12222 

3.0)244  4  —  2 

12 

124.4 

5)Z4 

4  —  4 
4 

^         5,16 

—  1 


120  B.ULE    OF    THREE    DIRECT. 

Ex.  2.  If  T  can  buy  27lb.  of  sugar  for  iL  13s.  how 
much  can  1  purchase  for  thirty  guineas  ? 
L,  s    lb.    guineas. 
1  13  :  27  : :  30 
20  21 

33  6J0 

27 


4410 
1260 
Ibsi 


33)17010(515 
165 

51 
33 

180 
165 

15 

16 
■  oz. 

33)240^7 
231 

9 
Ex.  5.  What  is  the  value  of  28  ells  of  cloth,  if  4  ells 

cost  18s.  ? 

ells,  shill.  ells. 

'  4  :  18  :  :  28 

18 


nULE    OF    THREE    DIRECT.  l2l 

Ex.  4.  If  six  yards  of  cloth  cost  24  shillings,  what  will 
SI  yards  cost?  Ans.   16/.  4s. 

Kx-  5.  If  8  gallons  of  wine  cost  17  dollars  25   cents,^ 
what  is  the  value  of  35  gallons  ?    Ans.  73  dols.  4b  cts,     ▼ 

Kx.  6.  If  5lb.  of  Dotatoes  cost  4ii.,  what  is  the  worth 
of  1  cwt.  on  the  samfe  terms  ?  Ans.  7.s.  5ld. 

Kx.  7.  If  5lb.  of  old  iron  cost  3rf.,  how  many  can  I  buy 
for  40>.  ?  Ans.  7  cwt.  0  qrs.  16  lb. 

Kx.  8.  If  10  English  ells  of  cloth  cost  11  dols.  11  cts., 
what  is  the  value  of  5  pieces,  each  containing  26  yards  ? 

Ans.  116  dollars  54  cents. 

Ex.  9  If  16  yards  of  muslin  cost  10  guineas,  how 
many  ells  can  I  buy  for  45^  Ans.  54  ells  4|S)q»'S. 

%    Ex.  10.  If  I  can  purchase  25  books  for  30  dollars,  how 
many  can  I  have  for  75  dols.  60  cts.  ?   .4ns.  63  books. 

Ex.  11.  If  a  servant's  wages' be  25  guineas  a  year, 
how  much  has  he  to  receive  for  87  days'  service  ? 

Ans.  6(,  5s.   lid.  ^. 

Ex.  12.  If  a  servant  receive  three  guineas  and  a  half 
for  20  weeks  service  :  how  long  ought  he  to  remain  in  his 
pliice  for  12  guineas.^  Ans.  68  weeks  4day8. 

tEx.   }J.  If  I  pay  half  a  crown  for  4Un  of  cheese  :  how 
much  can  i  have  for  three  crowns  and  nine-pence  ? 
"^  Ans.  25lb.  SSi  oz. 

Ex.  IJ^^  If  2  lb.  4  oz.  of  honey  cost  3>.  9rf. :  what 
is  the  value  of  28  lb.  ?  ^  \ns.  21.  6-.  8^, 

Ex.  yf.  If  3   lb.  of  sugar  cost  37a  cents,  what  will 
^  1  cwt.  amount  to  .?  Ans.   14  dollars. 

Ex.  1^  If  a  dozen  of  wine  glasses  cost  lOs.  6d.,  what 
is  the  value  of  dOO.P  Ans.  21/    \7s.  6d. 

•y    Ex.  \f.  If  I  can  buy  S  pair  of  shoes  for  7  dollars  50 
cts.  what  must  I  pay  for  17  pair  ?  Ans.  42  dols.  50  cts. 

Ex.  ^.  If  a  cwt.  of  tobacco  cost  8  guineas;  what  is 
the  value  of  7,000,000  of  ll)S.  ?  Ans.  525,000/. 

1^       Ex.   (9.  II  6  lb.  of  saop  cost  1  dollar  37a  cents,  what 
is  t!»e  value  of  1  cwt.  Ans.  25  dols.  66  cts. 

Ex.  10.  If  I  pay  39  shillings  per  cwt.  for  lead  ;  how 
much  will  it  cost  to  cover  the  roof  of  a  building  with  lead 
that  weighs  5505  lb.  >  Ans.  961,   \6s.  lid.  itz- 

Kx.  t|.  I  warit  to  know  how  much  1  have  to  pay  for  a 
cistern  990  lbs.  at  the  rate  of  2l.  2s,  per  cwt.,  the  plum- 
11 


Itt  RULE    OF    THREE    DIRECT. 

ber  agreeing  to  allow  me  at  the  rate  of  \l.  14s.  per  cwt: 
for  the  old  lead,  which  weighs  458  lb.  ? 
^  Ans.  11/.  I2s.  ^Id. 

^     Ex.  22.  If  a  journeyman  can  earn  9d*»llars  50  cents,  in 
6  days,  how  much  will  he  earn  in  3^)5  days  ? 

Ans.   4ro  dollars  9 1  cents. 

Ex.  23.  The  brazen  statue  of  Apollo,  that  was  erect- 
ed by  Chares,  at  Rhodes,  weighed  720,000  lbs.  :  how 
much  did  the  old  brass  sell  for  at  four  guineas  per  cwt.  I 

Ans.  27,000/. 

Ex.  24  If  I  pay  \L  7<i.  for  /8  gallons  of  porter,  how 
much  sh  II  1  expend  in  that  article  in  a  year,  if  my  fa- 
ffiily  drink  nine  gallons  of  it  every  week  ? 

Ans.  351.  2s,    ^ 

Ex.  25.  If  I  buy  14  gallons  of  brandy  for  35  dollars, 
Jjow  much  must  I  pay  for 4  hogsheads?  Ans.  630  dols. 

Ex.  £6.  If  I  buy,'  at  Sheffield,  6  razors  for  8s.  6d.  $ 
how  much  will  I  have  to  pay  for  twelve  dozen  at  the 
isaciie  rate  ?  And,  how  much  can  I  sell  them  fc^,  so  as 
to  gain  by  the  bat  gain  ^2ld,  on  each  razor  ? 

Ans.   10/.  4«.  cost,     1^  10.<;.  gain. 

Ex.  27.  In  building  an  out  house   5050   brii|cs  have  Jt 
been  used  ;   how   much  do  they  come  to  at  4s.  Od,  per 
huiitlred  p  Ans.  11/.  7v.  3d. 

Ex  28  It  requires  32  bricks  to  pave  9  square  (e*^t  : 
how  many  bricks  will  be  wanted  for  the  pavement  of  a 
«ellar  24  feet  lon^,  and  19  feet  wide    Ans.  16^  bricks.  ^ 

Kx.  29.  It  requires  144  Dutch  clinkers  to  pave  9  square^ 
(epf  :  how  many  will  be  wanted  for  a  court  35^et  long, 
and  29  feet  wi<le,  and  how  much  will  they  come  to  at  5s. 
6d.  ler  hundred  f      Ans.  16240  clinkers,  andj'^4t.  \3s^ 

Al        «0 

-•4     -lOO*  ^  . 

Ex.  30.  It  requires  sixty  persons  six  days  toxoanufac- 
ture  .  pack  of  wool  ij»to,  cloth  :  how  much  wool  will  they 
work  up  in  a  year,  supposing  they  work  6  daysiin  each  # 
week?  Ans.  433  packs. 

Ex.  31.  Six  children  of  different  ages  will  earn  in  five 
days,  at  spinning  wool,  5s.  9d..  and  the  mother  will  earn 
Ls  4</.  per  day  :  how  much  will  they  all  earn  in  a  vear, 
allowiji-^  thattUey  work,  one  week  with  anotheit^5*  tiays 
per  week  ?  Ans.  35/.  lOs.  '4d. 


RULB  OF    THREE  DIRECT.  1®3 

Ex.  32.  At  some  large  iron  founderies,  they  can   run 
•ff  6000  lbs.  of  iron  in  twenty-four  hours  :  how  many 
tons  weight  will  they  cast  in  a  year,  allowing  them  i%M 
work  298  days,  and  \Q  hours  each  day  ?  ^ 

f^  Ans.  5:i2  tons,  2  cvvt.  3  qr.  1?  lb. 

Ex.  33.  By  a  patent  machine  for  making  combs,  the 
teeth  of  two  combs  can  be  cut  in  thr^e  minutes  :  how 
many  can  be  manufactured  in  28  days,  if  the  machine  is 
worked  at  the  rate  of  eight  hours  a  day  ? 

Ans.  8960  combs. 
Ex.  34.  What  is  the  price  of  a  carpet  that  measures 
15  feet  each  way,  at  7s.  Qd.  for  every  9  square  feet  } 

Ans.  9^  7s.  Qd. 
Ex.  35.  If  13  cwt.  of  sugar  cost  150  dollars,  how  tsiuch 
must  I  pay  for  15  casks  of  the  same,  each   weighing  4 
cwt  2  qr.  12  lb.  ?  Ans.  797  dols.  39  cts. 

V^      Kx.  36.   How  much  flour  can  I  purchase  for  1087  dols. 
SO  cts.,  at  12  dols.  and  50  cts.  per  barrel  ^ 

Ans.  87  barrels. 
Ex.37.  If  candles  sell  for  2  dols.  12a  cts.  per  dozeii 
lb. ;  how  much  will  250  lb.  cost  ?     Ans.  44  dols.  27  cts. 
Ex.  38.  If  mould  candles  cost  l2.s.  6i.  per  dozen  :  how 
many  pounds  can  I  purchase  for  fifty  guineas.^ 

Ans.  84  dozen. 
Ex.  39  The  best  mottled  soap  is  bought  at  4^.  6s,  per 
cwt. :  for  how  much  must  it  be  sold   per  lb,,  so  as  to  al- 
low a  profit  of  one  penny  on  each  pound  ^ 

Ans.  lo^fi?.  nearly. 
Ex.  40.  If  I  buy  6^  yards  of  Irish  cloth  for  5  dols.  75 
cts. :  how  much  musi  I  pay  for  8  pieces,  each  cj)ntaiaing 
26  yards  r  Ans.  184  dols. 

^^  Ex.  41.  If  40  yards  of  Irish  cloth  will  make  12  shirts: 
how  many  may  be  made  out  of  4  pieces,  each  containg 
26  yards'?  Ans.  31^,  shirts. 

Ex.42.  If  12  gallons  of  brandy  pay  3  dol  ars  duty: 
V,     how  much  must  be  paid  for  37  hhds.   each  containing  63 
gallons  .P  Ans.  582  dols.  75  cts. 

Ex.  43.  The  average  price  of  sugar,  exclusive  ol  duty 
was,  Aug.  21,  1805,  M.  lis.  9\d.  per  cwt.  :  I  demand 
the  value  of  tie  9,'J99,360  lbs.  that  were  imported  into 
London  the  preceding  week  ?  Ans.  231105^ 


■]24  RULE    OF    THREE    DIRECT. 

Ex.  44.  The  average  price  of  tallow  was,  on  the  same 

day,  4s.  2d.  per  stone  of  8lb. :  what  is  the  worti)  of  276 

^ons,  imported  the  preceding  week?         Ans.  16.100^. 

^  Ex.^5.   What  will  31218  gallons  of  Port  wine,  sell 

for  at  7  dols.  1^2  cts.  per  dozen,  supposinjj;  each  dozen 

to  contain  3  gallons  ?  Ans.  74142  dols.  75  cts 

Ex.  46.  \\  hat  is  the  value  of  115  seal-skins,  at  3s.  6d. 
per  lb.  supposing  the  skins  to  weigh,  one  with  the  other, 
0  ounces  each  ?  Ans.  I  It.  6s.  4ldfQ. 

Kx.  47.  How  many  quills  can  I  have  for  156  dols.  25 
cts.  at  622  cts.  per  hundred  ?  Ans.  25,000 

Ex.  48.  How  much  brown  Holland  can  I  buy  for  ten 
guineas,  if  1  pay  5s.  9d,  for  four  yards  and  a  quarter  ? 

Ans.  155a^3  yards. 

Ex.  49  Suppose  a  person  save  out  of  his  income  5s. 
%d-  per  week  :  how  long  will  he  be  laying  by  100/. 

Ans.  363  weeks  4^  days  nearly. 

Ex.  50.  I  want  to  know  the  height  of  a  tree,  by  means 
of  the  length  of  its  shadow  ;  1  set  up  a  straight  stick 
that  measures,  above  the  ground  3  feet  4  inches  ;  the 
shadow  of  this  is  5  feet  2  inches,  and  the  shadow  of  the 
tree,  at  the  same  moment  I  find  to  be  79  feet  10  inches  ? 

Ans.  51  leet,  6^. 

Ex.  51.  What  is  the  height  of  a  steeple,  whose  sha- 
dow is  148  feet  4  inches,  when  a  shadow  5  feet  3  inches 
long  is  projected  from  a  staff  6  feet  4  inches  ? 

Ans.   178  feet,  11^. 

Ex.  52.  If  I  pay  4s.  9d.  for  a  hundred  of  pens  :  how 
many  shall  1  get  f»V  lOO/.  Ans.  42I05i*9. 

Ex.  53.  If  my  income  is  450/.  per  ann.  :  how  much 
may  I  spend  in  73  days,  supposing  I  mean  to  lay  by  5a 
guinea*  at  the  year's  end  }  Ans.  79/   10s. 

Ex  54.  What  is  the  value  of  57  yards  of  inuslin,  at 
the  rate  of  8  7^cts.  per  Ell  English  ^  Ans.  39  dols  90  cts. 

Ex.  55.  How  many  Ells  English  of  cloth  can  be  bought 
for  73  dols.  87a  cts. :  at  the  rate  of  1  dollar  50  cts  per 
yanJ  i  Ans.  29  E.  E.  2  qrs. 

Ex  56.  1  have  a  tankard  that  weighs  2  lb.  3  oz.  that 
cost  H)/.  2s.  6d.  ;  how  much,  at  the  same  rate,  will  a 
service  of  plate  cost  that  weighs  l23lh.  9oz.  ? 

Ans.  o%5i.  irs.  6d. 


RULE    OF    THREE      DIRECT  125 

Ex.  57.  How  much  must  be  paid  for  74  yards  of  cloth, 
at  372  cts.  per  Kll  Flemish  ?  Ans.  37  dolsi 

Ex.  58.  What  was  paid  for  22,275  bushels  of  flour,  at 
4^  7>t.  the  sack  of  five  bushels  ?  Ans.  19,379/.  5s. 

Ex.  59.  What  is  the  value  of  6  casks  of  raisins,  each 
weigliing  3  cwt.  2  qrs.  14  lb.  at  31.  \0s   6(1.  per  cwt.  ? 

Ans.  1 201.  Ss.  4*rf. 
Ex.  60.  How  much  must  I  jrive  for  a  gold  snutt-box 
that  weighs  S  oz.  9  dwts.,  at  the  rate  of  4/.  3s.  9ci.  per 
oz  }  Ans.  35/.  7.s.  S^c/. 

Ex.  61.  How  much  tax  must  I  pay  for  586  dollars,  at 
61  cts.  per  dollar  ?  Ans.  36  dols.  62^  cts. 

Ex.  6'2.  A  Bankrupt  has  but  1020  dollars  to  pay  debts 
to  the  amount  of  3235  dollars  ;  how  much  can  he  pay  in 
the  dollar  }  Ans.  31  cts.  7  mills  nearly. 

Ex.  63.  A  merchant  failing,  his  assignees  find  eflfects 
and  good  debts  to  the  aniount  of  3335/. ;  but  he  owes 
4225 1. ;  the  expences  attending  his  bankruptcy  will  be 
212/.  9s.:  how  much,  therefore,  will  he  pay  in  the 
pound.?  Ans.  l4i»/9c?.S 

Kx.  64.  An  honest  tradesman,  tlirough  unfore-eei> 
misfortunes,  is  obli^jed  to  call  his  creditors  togetlier  ;  he 
,fin«ls  his  debts  to  be  432b/.  and  he  can  pay  J4.s.  6d.  \n 
the  pound  :  how  much  has  he  still  left  ?  Ans  3136/  7s. 
Kx.  65.  Hops  are  remarkably  cheap,  and  I  have  100/. 
to  spare  :  how  many  can  [  purchase  at  3/.  15s  6//.  per 
cwt.  Ans.  26  cwt.  2  qrs.  nearly. 

Ex.66.  If  10  lbs.  of  tea  are  worth  10  dols.  25  cts.; 
how  much  of  the  same  sort  can  1  purchase  for  38  dols. 
431  cts.  ?  An.«.  37  lb.  8  oz. 

Ex.  67.  What  must  I  pay  for  the  carrisig**  by  th«^'.  ca- 
nal, froui  Manchester  to  Etruri;i,  705  tons*  5  cwt.  of 
goods,  at  15s.  per  ton  ;  and  what  is  the  diHeren-  e  be- 
tween tlus  and  the  ia'id  caniage,  at  "Zl.  I5s.  per  ton  ? 

Ans.  I  iiust  pay  62s/,  I8>.  9^/..  for  carriage  by  ('anal^ 
and  14-10/,  10s.  oittV'FtM.ce  in  t  :e  price  of  carriage. 

Ex  68.  \\h.iit  weight  of  goods  can  be  r;i'7;'"d  on  the 
the  canal  l»etv\ern  Manchester  and  Hjrniiitjjf^^-:i  f(»r  85/- 
at  the  rate  of  I/.  10s.  per  ton  :  and  liow  lo  ^.p  fan  be 
earned  the  same  distance,  by  land-fHrria;^e,  at  5/.  per 
toa  i      Ana*  6G|  tons  by  water,  and  17  tons  by  land. 


126  RULE  OF  THREE    DIRECT.. 

Ex.  69.  The  clothino;  of  a  regiment  of  760  men  cornea 
to  3050/  how  much  is  that  per  man  ?  Ans.  4/. 7^. 

Ex.  70.  What  may  a  man  spend  per  week,  whose  in- 
come is  2000/.  per  annum,  supposing  52  weeks  in  a 
year  ?  Ans.  38/.  9.s.  2i;/.^a. 

Ex.  71.  If  hy  selling  fine  Irish  cloth  at  5  dollars  per 
ell,  I  gain  108  dollars  ;  how  much  shall  I  gain  if  I  sell  it 
at  6  dols.  25  cts.  per  ell  ?  Ans.  135    ollars. 

Ex.  72.  If  sugar  that  cost  9  cts.  per  lb.  be  sold  at  3  lb. 
for  37a  cts.  what  will  be  the  gain  by  selling  1  cwt>  ? 

Ans  3  dols.  P2  cts. 

Ex.  73.  I  purchase  5  pieces  of  Holland,  each  contain- 
ing 30  yards,  at  4s.  9c?.  per  yard  :  how  much  shall  I  gain 
by  selling  it  at  6s.  2rf.  per  ell  English  ? 

.Ans.  I/.  13s.  profit. 

Ex  74.  Two  persons  part  at  the  same  time  for  the 
same  place,  tlie  one  travels  north  24  miles  a  iVdj,  and 
the  other  21  miles  a  day  south  :  when  w\\\  they  be  1000 
miles  asunder  ?  Ans.  224  days  nearly. 

Fix.  76.  If  a  pack  of  wool  weighs  3  cwt.  2  qrs.  7  lb., 
what  is  it  worth  at  2 Is.  bcl,  per  tod  of  14  lbs.  ? 

Ans.  30/.  \2s.9d, 

Ex.  76.  The  rents  of  a  parish  amount  to  1750/.,  and  a 
rate  for  the  poor  is  wanted  of  65/.  7s.  6d* :  wliat  is  that 
per  pound?  Ans.  9rf.  nearly. 


(   127  ) 


THE  RULE  OF  THREE  INVERSE. 


This  rulp,  like  the  last,  teaches,  from  three  given 
numbers,  to  find  a  fourth,  which  fourth  number  shall 
bear  the  same  proportion  to  the  second,  as  the  first  has 
to  the  third.  'I'hus,  if  the  question  be,  .If  10  men  can 
nmw  a  certain  field  in  6  days,  how  soon  can  it  be  done 
by  2U  n»en  ?  The  answer  will  evidently  be  in  3  days, 
because  do  ibie  the  number  of  men  will  certainly  do  the 
same  work  in  half  the  time;  the  proportion  will  therefore 
stand,  10  men  :  0  days  :  ;  20  men  :  3  days  ;  and  3  bears 
the  same  proportion  to  &.  that  10  does  to  30  ;  that  is,  the 
fourth  number  bears  the  same  pri>portion  to  the  second, 
that  the  first  does  to  the  third. 

Rule. — State  the  question,  and  when  necessary,  re- 
duce the  terms  as  before.  Multiply  the  first  and  s*eco)id 
terms  together,  and  divide  the  product  by  the  thud 
term  ;  the  quotient  is  the  answer  in  the  same  denomina- 
tion as  the  second  term  5  thus  in  the  foregoing  example> 

JOxt) 
IZ3  days. 

20 

Ex.  1.  If  15  reapers  can  cut  down  a  field  of  corn  in  4 
days,  in  how  long  time  will  the  same  work  be  performed 
by  40  men  ? 

15  :  4  :  :  40 
4- 


4.0)0.0 


128  RULE    ©F    THREE    INVERSE* 

Ex.  2.  If  the  penny  loaf  weighs  4  ounces  when  flour 
is  4.S,  per  peck,  how  much  must  it  weigh  when  flour  is 
5s.  4>d.  per  peck  ?  Ans.  3  ounces. 

Ex.  3.  A  person  lent  me  240  dollars  for  8  months  ; 
in  return  for  his  kindness,  how  much  ought  I  to  lend 
him  for  18  months  ?  Ans.   106  dollars  66  cents. 

Ex.  4  How  many  men  must  be  employed  to  finish  a 
canal  in  12  days,  which  5  could  perform  in  six  weeks,  or 
36  days  ?  Ans.   15  «nen. 

Ex.  5.  If  24-  pioneers  can  make  a  trench  in  12  days, 
what  length  of  time  would  the  same  work  employ  9  men  P 

Ans.  3-2  days. 

Ex.  6.  The.  floor  of  a  chapel  96  feet  in  length,  and  7Q 
feet  in  breadth,  is  to  be  covered  with  matting  2  feet  six 
inches  broad  :  how  many  yards  will  it  require  ? 

Ans.  2688  feet 

Ex.  7.  If  a  person  travel  12  hours  a  day,  and  finish 
his  journey  in  three  weeks  ;  how  hmg  would  the  same 
journey  take  bin),  if  he  travelled  only  9  hours  a  day  at 
the  same  rate  ?  Ans.  4  weeks. 

Ex.  8.  If  the  town  and  garrison  of  Bhurtport,  con- 
taining 22.400  persons,  have  provisions  to  last  three 
weeks,  how  many  tnhaiiitants  must  tiolkar  send  away, 
so  as  to  make  the  provisions  last  7  weeks,  which  is  as 
long  as  General  Lake  can  carry  on  the  siege  ? 

Ans.   12,8oo  persons. 

Ex.  9  If  a  besieged  garrison  have  4  months  provisions, 
at  the  rateot  18  ounces  per  man  per  day  ;  how  long  wdl 
they  be  able  to  hold  out,  if  each  man  is  allowed  only  \Z 
ounces  per  day  ?  Ans.  6  months. 

Ex.  10.  If  there  arc  in  a  garrison  provisions  sufficient 
for  150(/  men  10  weeks,  which,  on  account  of  the  rains, 
is  seven  weeks  longer  tiian  the  siege  can  last;  how  many 
sohhers  may  be  brought  to  defenti  the  place  for  three 
weeks,  without  lessening  the  quantity  of  food  tt>  anj 
individual  ?  ^    Ans    330J  soldiers. 

Ex  11  If  9  plasterers  can  finish  the  uiside  ol  a  cha- 
pel in  10  days:  how  lonjj  wdl  it  take  4  rt>en.  supoo-in* 
the  other  3  sent  away  '-n  a  i»ew  job  P      Ans.   22a  *'•*>*• 

Ex.  12  it  S3  yards  of  broadcloth,  if  wide  will  usake 
a  suit  of  clotlies  ;  how  xuuch  will  be  nece^sai  v  of  cloth 
onlj-  I  wide  ?  Ans.  8  jards  0|. 


^  RULE    OF    THREF:    INVERSE.  1 

Kx.  13.  If  32  clerks  in  the  bank  are  sufficient  to  make 
up  the  books  in  a  certain  office  in  15  days,  how  many 
clerks  would  be  required  to  do  the  same  work  in  6  days  ? 

Ans.  130  clerks. 

Ex  14  If  the  carriage  of  15a  cwt.  for  60  miles,  came 
to  7s.  9d.  ;  how  far  can  [  have  carried  3^  cwt.  for  the 
same  sum  }  Ans.   248  miles- 

F^x.  15.  If  24  men  can  finish  a  piece  of  w>)rk  in  \d 
hours;  how  manv  men  will  it  require  to  do  the  same 
work  in  12  hours?  Ans.  32  men. 

F.x.  ^6.  If  12  inches  in  length,  and  1^  inches  in 
breadth,  njake  a  square  foot ;  what  length  of  board,  8 
inches  broad,  will  be  equal  to  the  same  measure. 

Ans.   18  inches. 

Ex.  17.  If  220  yards  in  length,  and  22  in  breadth, 
make  an  acre  ;  what  must  be  the  breadth  when  the  length 
is  120  yards  }  Ans.  40  yards. 

Ex.  18.  If  5  horses  can  he  maintained  when  o<its  are 
18.S.  per  quarter;  how  mar)y  can  be  supported  at  the 
same  cost,  when  they  are  30  shillings  per  quarter  } 

Ans.  3  horses. 

Ex.  IP.  If  2J0  dollars  gain  12  dollars  at  interest  in 
12  months  ;  what  principal  will  gain  an  equal  sum  in  5 
months  .f^  Ans.  60()  dollars. 

Ex.  20.  There  are  two  rooms  in  the  floors  of  which 
there  are  an  fqiial  number  of  square  feet ;  the  length  of 
the  one  is  50  teot,  af;d  its  breadtf^is  42  :  hut  the  breadth 
of  the  other  is  48  f*-et;  what  is  its  length  ?     Ans   43^. 

Kx  21.  The  cock  to  a  large  water-tub  will  eii.pty  it 
in  S>j  minutes  ;  how  many  such  cocks  will  empty  it  in 
42  ndnutes  .''  Ans.  8  cocks. 

Ex.  22.  The  sides  of  a  room  are  found  to  measure  J  38 
feet  in  lenu;th.  a')d  thf  height  is  14  feet  6  inches  ;  how 
much  paper,  2  feet  3  inches  wide,  will  cover  it  ;  and 
what  is  the  valun  of  it  at  9rf.  per  yard  } 

Ans.  296  yds.  I  ft.  4  in.     \\l.  9s    4d. 

Ex.  23  If  50  cows  can  be  kept  in  a  field  17  days; 
how  Ion";  will  the  same  pasture  feed  70  cows  ? 

Ans.   125. 


(  no  ) 


THE  DOUBLE  RULE  OF  THREE. 


THE  Double  Rule  of  Three  teaches,  from  five   given 
nuuihers  to  find  a  sixth.     Three  of  the  numbers^ontain     - 
the  suppositions,  and  the  remaining  two  are  terms  of  de- 
mand. 

Rule  (1)     Put  tlie  terms  of  supposition  one  above 
another  in  tlie  first  place,  except  that  which  is  of  the 
same  nature  with  the  term  sought,  which  put  in  the  se-     a 
cond  place. 

(2.)  Place  the  terms  of  demand  one  above  another  in 
the  third  place,  in  the  same  order  as  the  terms  of  the 
supposition  were  put  in  the  first  place. 

(3.)  The  first  and  third  term  in  every  row  will  be  of 
the  same  nature,  and  must  be  reduced  to  one  denomina- 
tion ;  and  the  middle  term  must  be  brought  to  the  low- 
est denomination  mentioned. 

(4.)  Examine  each  stating  separately,  using  the  mid-    w 
die  term  as   common  to  both,  in  order  to  know  if  the 
proportion   be   direct  or   inverse.     When    it  is   direct 
mark  the  first  term  with  an  asterisk,  and  when  it  is  in- 
verse, mark  the  third  term  with  an  asterisk. 

{5.)  Multiply  the  numbers  together  which  are  marked 
for  a  <livisor,  and  those  which  are  not  marked  for  a  divi?   ^ 
dend,  and  the  quotient  will  be  the  answer. 


DOUBLE    RULE    OF    THUEK.  131 


Ex.  1.    If  12  persons  spend  160^in  4  months  :  how 
much  will  32  persons  expend  in  8  months'? 


persons. 

L. 

persons. 

1-2    : 

I(j0 

:  :  32 

months 

months. 

4     : 

or, 

:  :     8 

12  X  4  : 

160  ; 

s  :  S2  X  8 

32  X 

:  8  X 

160 

=ZS53L 

6s, 

8rf. 

12  X  4 

Ex.  2    If  a  garrison  of  600  men  have  provisions  for  5 
weeks,  al  owiny;  each  man  12  ounces  per  day  ;  how  many 
can  be  maintained  10   w^-.eks  by   the  same   quantity,  if 
^     each  man  is  limited  to  8  ounces  a  day  P 

weeks,     men.     weeks. 

5     :     600     :  :     10 

oz.  oz, 

12     :  !  *•     8 

or, 

5X  12  :  600  ::  10  x  8 

6  X   12  X  600 

-"450  men,  Ans. 

10  X  8  ^ 

Ex.  1.  If  15  pecks  of  wheat  will  last  a  family  of  P 
perso">  22  da  vs.  in  how  many  days  will  six  persons  coR- 
sume  20  pecks  ? 


pecks. 
15     : 

days. 

22     : 

pecks. 
;     20 

persons 

9     : 

persons. 
: :  6 

9  : 

or, 

^  21  X  20 

— ^ZZ  44  days' 

15  X  6 

132  DOUBLE  RULE  OF  THREE. 

Ex.  4.  If  6  pioneers  can  dig  a  ditch  34  yards  long  in 
10  davs  ;  how  many  yards  may  be  dug  by  20  men  in  15 
days  ?  Ans.   170  yards. 

Ex.  5.  If  1050  soldiers  consume  230  quarters  of  corn 
in  6  months ;  how  many  soldiers  will  960  quarters  serve 
4  months  ?  '  Ans    6048  men. 

Ex.  6.  If  a  cask  of  beer  last  8  persons  14  days  :  how 
man)'  casks  will  serve  2  persons  3()5  days  ? 

Ans.  6^  casks. 

Ex  7.  If  10  men  in  6  weeks  earn  500  dollars  ;  how 
many  weeks  must  15  men  work  to  earn  lOOO  dols. 

Answer,  8  weeks. 

Ex  8.  Suppose  I  walk  66  miles  in  4  days,  of  eight 
hours  each  day  :  how  n)any  days,  of  14  hours  each,  shall 
I  be  in  going  from  London  to  York,  or  196  mites. 

Ans.  6^.  almost  7  days. 

Ex.9.  If  three  boats  take  6000  herrings  in  8  days  : 
how  long  will  (iOO  boats  be  in  taking  20,000  barrels,  each 
containintj;  700  herrings  ^  Ans.  ^Sg 

Ex.  10.  If,  against  a  general  mourning,  6  tailors  can 
make  JO  suits  of  clothes  in  4  days  :  how  many  suits  can 
600  men  make  in  the  7  days  which  occur  before  the 
mourning  is  wanted  ?  Ans.  1750  suits, 

Ex.  11.  If  12  niantua-makers  can  make  27  mourning 
dresses  in  4  days  :  how  many  persons  would  be  required 
to  make  189  dresses  in  8  days  ? 

Ans.  42  mantua-makers. 

Ex.  12  If  3000  copies  of  a  History  of  America,  each 
containing  U  sheets,  require  66  reams  of  paper  :  how 
much  paper  will  5000  take,  if  the  work  be  extended  t» 
123  sheets.  Ans.  125  reams 

Ex.  13.  As  12  inches  in  length,  12  in  breadth,  and 
12  in  thickness,  make  a  solid  foot :  what  length  of  plank, 
which  is  7  inches  broad  and  3  inches  thick  will  make  the 
same?  Ans.  827 inches. 

Ex  14.  If  430  tiles,  each  12  inches  square,  will  pave 
my  cellar  :  how  many  tiles  must  1  have,  if  the  tiles  are 
9  inches  long  and  8  broad  ?  Ans.  900  tiles. 

fcx.  15.  If  the  expence  of  3  persons  on  a  tour  for  5 
months  be  133/.  8s.  :  what  will  2  persons  spend  in  9 
months  ?  Ans.  148^  Is,  Id. 


MISCELLANKOUS  QUESTIONS.  IS3 

Ex.  16.  If  12  ounces  of  wool  make  2l  yards  of  very 
tine  cloth,  6  quarters  wide :  how  much  wool  would  be 
required  to  loO  yards,  4  quarters  broad  ? 

Ans.  480  ounces. 
Ex.  17.  If  300  dollars  gain  15  dollars  interest  in  a 
year,  in  what  time  will  900  dollars  gain  180  dollars. 

Ans.  4  years. 
Ex.  18.  If  an  iron  bar  4feetlong,  3  inches  broad,  and 
IJinch  thick,  weigh  36  lbs.  :  how  much   will  a  bar  weigh 
that  ts  SIX  feet  long,  4  inches  broad,  and  2  inches  thick  > 

Ans.  ll5Hbs. 


MISCELLANEOUS    QUESTIONS    ON    ALL     THE    FOUEGOING 
RULES. 

Ex.  1.  What  three  numbers  are  those,  the  first  of 
which  is  105,  the  second  ^ds  of  the  first,  and  the  third 
67  less  than  the  first  and  second  together  I 

Ans.  first,  105,  second  70,  third,  108. 

Ex.  2.  A  gentleman  left  his  eldest  daughter  1000  gui- 
neas more  than  the  youngest,  and  to  tliree  other  daugh- 
ters he  left  7000  guineas  between  them,  which  was  equal 
to  the  sum  left  to  the  youngest  and  eldest  together  : 
what  was  each  child's  fortune.  Answer,  Eldest,  4000, 
youngest,  3000,  three  other  daughters,  7000. 

Ex.  3.  W  hat  is  the  difference  In  value  between  five 
times  five  and  twenty  guineas,  and  five  times  twenty- 
five  guineas  ?  Ans.  80  guineas. 

Kx.  4.  What  was  the  value  of  a  prize  taken  by  25  sai- 
lors, besides  officers,  so  that  each  sailor  received  19^.  9s, 
96/.,  and  the  officers  received  as  much  as  the  sailors  ? 

Ans.  974/.  7s.  6d, 

Ex.  5,  A  prize  valued  at  13,177/.  10s.,  after  the  offi- 
cers have  had  their  share,  is  to  be  divided  among  525 
sailors  :  what  would  each  man  have  to  take  ? 

Ans.  25/.  2s.  each  man's  share. 

Ex.  6.  What  is  a  fourth  proportional  to  the  numbers 
6,  9,  and  24  ?  Ans.  36  fourth  proportional. 

Ex.  7.  What  is  the  value  of  4  packs  of  cloth,  eack 
pack  coutainiug  4  parcels,  each  parcel   10  pieces,  and 
12 


134  MISCELLANEOUS  QUESTIONS. 

each  piece  £6  yards,  at  the  rate  of   12  dols.  50  cts.  for 
3  vards  ?  Ans.  17333  dols.  33  cts. 

Ex.  8-  How  many  yards  of  paper,  3  quarters  wide, 
will  be  sufficient  for  a  room  48  yards  round,  arid  four 
yards  high  :  and  what  is  the  value  of  the  paper,  at  the 
rate  of  18s.  per  piece  of  24  yards  ? 

Answer.  2o6  yards,  worth  9/.  12s. 

Ex.  9.  If  100  dollars  jiain  5  dollars  in  J 2  months, 
what  will  75  dollars  gain  in  9  months  ^ 

Ans,  2  dols.  81^  cts. 

Ex.  10.  If  48  cannon  consume,  in  3  da;  s,  288  barrels 
of  powder,  how  much  will  be  spent  in  15  days,  when 
144  cannon  are  to  be  supplied  ?  Ans.  4320. 

Ex.  ll.  Fifteen  people  joined  to  purchase  a  lottery 
ticket,  for  which  they  gave  three  sshillings  less  than 
eighteen  guineas:  if  it  came  up  a  prize  of  30,000  gui- 
neas, what  did  each  n  an  receive,  and  what  was  his 
gain  }  Ans.  2100/.  each  man's  share,  and  2098/.  15s. 
each  man's  gain. 

Ex.  12.  A  tobacconist  bought  two  parcels  of  tobacco, 
which  weighed  9  cwt,  2  qrs  ,  for  a  hundred  guineas,  the 
difference  of  the  parcels  in  weight  was  3  qrs.  12  Id.,  and 
in  value  eight  guint'as  :  what  was  their  weight  and  va- 
lues ?  Ans.  one  parcel.  5  cwt.  Oqr.  20  lb.  the  other  par- 
cel 4  cwt.  I  qr.  8  lb.,  cost  54  guineas,  and  36  guineas. 

Ex.  13.  'J  he  clothing  of  100  charity  children  came  to 
211/.,  of  which  135/.  was  expended  on  60  boys:  what 
was  paid  for  the  40  girls,  and  how  much  did  the  clothes 
of  each  child  cost?  Ans  gris  clothing  76/.,  price  of 
each  boy's  clothes,  2/.  5s.  ditto  girl's  clothes  1/.  18s. 

Ex.  14.  A  great  graz-ier  left  to  his  four  sons  220  oxen 
and  1200  ^heep  :  1  demand  the  value  of  each  son's  lega- 
cy, supposing  the  oxen  worth  18  guineas  each,  and  the 
sheep  39  shillings  each  ?  Ans.  1624/.  10s.  each  sou's 
legacy. 

15.  What  number  is  that  which,  multiplied  by  384, 
will  give  a  product  of  3.013,2+8  ?  '  Answer,  7847 

Ex.  16.  What  is  gained  by  the  sale  of  456  yi  rds  of 
cli)th.  that  was  bought  at  the  rate  of  7  dols.  25  cts.  per 
yard,  and  sold  at  the  rate  of  11  dols.  50  ct^.  per  yard  ? 

Ans.  1853  dols. 


MISCELLANEOUS  qUESTrONS.  135 

Ex.  17.  ir  9  printers  can  set  up  the  New  Testament 
in  %%  (lays,  in  what  time  could  it  be  done  if  15  were  em- 
ployed ?"  Answer,   1  Sg  day^s. 

Kx.  18.  If  8  men  will  earn  on  an  average  84  dols.  in 
6  days,  how  much  can  16  men  earn  in  27  days  ? 

Ans.  708  dols.  75  cts. 

Ex.  19.  When  the  quotient  is  1083,  and  the  divisor 
555,  what  is  the  dividend,  if  there  be  a  remainder  of 
79?  Answer,  601144. 

Ex.  20.  The  silk  mil!  at  Derby  winds  oft' 7 3.726  yards 
of  silk  every  time  the  great  wheel  goes  round,  which  is 
thrice  in  a  minute  :  how  many  yards  wdl  it  wind  in  a 
year,  allowing  that  it  works  every  day,  except  Sunday, 
15  hours,  and  how  many  skeins  will  be  made,  supposing 
960  yards  gO  to  the  skein  i*  Ans.  62,305,842,(iOO  yards 
made  in  a  year,  and  64901919s  the  number  of  skeins. 

Ex.  21.  In  tlie  partition  of  some  waste  lands  in  the 
west  of  England,  A  had  59^  acres,  B  761  acres,  C  110 
acr.  2r.  12  per.,  D.  15  acres,  and  E  39  acr.  Or.  12  n-r.. 
but  these,  taken  together,  were  but  one-filth  of  the  whole  : 
how  many  acres  were  divided,  and  what  was  the  value 
of  the  whole,  supposing  each  acre  worth  16^.  9s.  6rf.  } 
Ans.  1502  acr.  .Or.  Op.  land  divided,  23243/.  9s.  Oc/.  va- 
lue of  the  land. 

Ex.  22.  An  Island  in  the  West  Indies  contains  42 
parishes,  and  every  parish  7Q  houses,  and  each  house  at 
thtj  rate  of  b\  white  persohs  ;  besides  these,  there  were 
65  negroes  to  each  of  54  plantations  :  how  many  people 
were  there  on  the  whole  island  ?     Ans.  21066  persons. 

Ex.  23.  \n  the  club  mentioned  in  the  Spectator 
(No.  9.),  there  were  15  persons,  weighing  together  3 
tons  :  how  many  pounds,  ounces,  and  diains.  Avoirdu- 
pois, did  each  man  weigh  .'^  Ans.  448  lb.  7168  oz. 
ll468Bdr. 

Ex,  24.  The  British  possessions  in  Hindostan  contain 
212,406  square  miles,  and  the  population  is  estimated  at 
fourteen  miilions  :  how  many  inhabitants  are  tiiere  to  a 
square  mile  }  Ans.  66  persons  nearly, 

Ex.  25.  If  9  lb.  of  tea  cost  7  dollars  20  cts.,  what  is 
the  worth  of  4  chests  each  weighing  l  cwt.  2  qi  s.  ? 

Ans.  537  dollars  60  cents. 

*  365—52  equal  313  the  number  of  working  days  in  a  year. 


136  mSOELLANEOVS    QUESTIONS. 

Ex.  26.  What  shall  I  give  for  a  farm  containing:  256 
acres,  for  which  I  am  to  pay  at  the  rate  of  95  dollarii  for 
4  a^res  ?  Ans.  6080  dollars. 

Ex.  27.  What  will  it  cost  a  young  man  to  come  into  a 
farm,  for  the  lease  of  which  he  is  to  pay  lOOO  guineas; 
fur  22  horses  he  is  to  pay  at  the  rate  of  18  guineas  each  5 
for  crops  in  t'  e  ground  354/.  ;  for  210  hushels  of  wheat 
he  is  to  pay  4l.  los.  per  8  bushels  ;  the  household  furni- 
ture is  appraised  to  him  at  298  guineas,  and  for  farming, 
utensils  of  all  kinds  he  is  to  pay  196/.  ? 

Ans.  2i46/.   16.s.  6d.  ^ 

Ex.  28  The  revenue  collected  in  Hindostan  by  the  Bri- 
tish, is  reckoned  at  3,400,000/.,  how  much  is  that  from 
each  inhabitant,  supposing  they  amount  to  14- millions? 

Ans".  4s.   \Old. 

Ex.  29.  The  number  of  negroes  in  Jamaica  is  estimat- 
ed at  250,000,  and  of  whites  20,000,  how  many  slaves 
are  there  to  a  single  white  man,  and  what  do  the  planters 
:-.-.'I...;.  their  nronprfv  i.nr+u  \n  the  article  of  slaves  only, 

supposing  each  to  be  worth  93  guineas  r 

Ans.  125  slaves,  24,412,500/. 

Ex.  30.  The  population  of  the  United  States  is  esti- 
mated at  six  millions  and  a  half,  and  the  number  of  slaves 
still  existing  in  that  free  country  is  reckoned  to  be 
697,097,  how  many  free  people  are  there  to  one  slave  ? 

Ans.  93  nearly. 

Ex.  31.  The  extent  of  China  Proper  is  equal* to 
1,397,999  square  miles,  and  the  population  is  estimated 
at  333,000,000,  how  many  inhabitants  are  there  to  a 
square  mile  .^  Ans.  238  nearly. 

Ex.  32.  In  Spain  each  person  pays  10  shillings  to  go- 
vernment for  protection  ;  in  France,  under  tlie  old  go- 
vernment, each  paid  20s.  for  protection  ;  and  in  England 
we  pay  full  three  guineas  each  for  the  same  advantages, 
how  much  is  the  revenue  of  the  three  governments,  suppo- 
sing the  population  of  Spain  to  be  10^  millions ;  of  France, 
at  the  period  referred  to,  25  millions  ;  and  of  England 
and  Wides  9,343,1 73.?  Ans.  59,555,994/.  19s. 

Ex.  33.  The  population  of  London,  Westminster, 
and  fcouthwark,  is  8()4,8d5,  that  of  Paris  547,75t),  how 


MISCELLANEOUS    qUESTlONS.  137 

much  does   the  population  of  London  exceed   that  of 
Paris?  Ans.  317.109. 

Kx.  34.  How  many  minutes  and  seconds  have  elaps- 
ed since  the  birth  of  Christ,  or  1808  vears*  ? 

Ans.  95'),Q35,6RO  min.     57,056,140,800  sec. 

Ex.  35.  How  Ions;  would  it  require  to  count  five  hun- 
dred tnillions  sterling,  supposins;  a  person  were  to  reck- 
on 150L  in  a  minute,  and  were  to  he  employed  10  hours 
ea  h  day,  and  six  days  a  week,  till  he  had  finished  the 
job?  *  Ans.  9^2(>  weeks,  nearly. 

Kx.  3d.  How  many  barley-corns  will  reach  round  the 
earth,  supposing  the  length  to  be  25,200  miles  ? 

Ans.  4,790,016,000 

Ex.  37.  How  many  seven-slnHinw  pieces  are  there  in 
a  thousiind  pounds  ?  Ans.  28.57  seven -shil.   1  shil. 

Ex.  38.  A  French  franc  is  worth  \0d.,  how  many 
francs  are  there  in  100^  Ans.  2400  Francs. 

Ex.  39.  If  8  men  can  mow  18  acres  in  4  days,  how 
many  men  will  be  required  to  mow  50  acres  in  six 
days  f  Ans.  14f^ 

kx.  40.  A  balloon  has  moved  at  the  rate  of  6492  feet 
in  a  minute,  how  long  would  it  have  been  sailing  round 
the  earth  at  the  same  rate,  supposing  the  circumference 
of  the  earth  to  be  ^5,^^200  miles  } 

Ans.   14  days  5  hours  35  min,  22  sec. 

Ex.  41.  How  much  oftener  will  the  small  wheel  of  a 
coach  turn  than  the  large  «>ne,  between  London  and  Bris- 
tol, or  120  miles,  if  the  former  be  10  feet  8  inches  in 
circumference,  and  the  latter  18  feet  4  inches  ? 

Ans.  24840. 

Ex.  42.  If  my  income  be  250^  per  annum,  and  I  have 
fo(»lishIy  expended  15.s.  per  day,  how  much  <hail  I  be  in 
debt  at  the  year*-  eod,  and  what  may  I  expend  per  <l.iy  the 
following;  year,  so  as  to  have  ten  guineas  in  hand  at  the 
conclusion  of  it?    \ns  2.3/.  15s.  debt,   lis.  911    speid. 

Ex.  43.  It  i*  said  the  impositions  of  hackney  coach- 
men, by  overcharges,  are  equal  to  one-fiurth  of  what 
they  earn  ;  now,  if  ibev  earn  each  on  an  average   IS.*. 

r;r  day,  and  there  be  1 100  employed  31 J  ;lays  in  a  year, 
demand  tiie  amount  of  their  overcharges  in  a  year  ? 

Ans.  7 7 467 i.   lOS. 
♦  Allowing'  365^  days  in  one  year. 


138  MISCELLANEOUS    qUESTIONS. 

Ex.  44.  There  were  at  Vauxhall  ajardens  on  fhe 
Prince  of  Wales  birth-day,  1805,  10,059  persons:  the 
admission  money  wa-i  Ss.  each;  now,  supposinsc  oach 
person  to  spend  3s.  more,  the  half  of  which  was  r>»'ofit 
to  the  proprietor,  what  would  he  clear  hy  the  night,  al- 
lowing that  the  incidental  expenses  were  250/.  ? 

Ans.    1258/.   17s. 
Ex.  45.  If  3l  yards  of  cloth  will  make  a  shirt,  how 
much  of  the  same  stuft' will  he  wanted  to  make  two  shirts 
for  each  man  of  a  regiment,  consisting  of  855  men  ? 

Ans.  555/3  yards. 
Ex.  46.  In  November,  1800,  276,334  five-pound  hank 
notes  were  issued  ;  in  l)ece;nhcr  2,626,700  ;  and  in  the 
January  following  2,769,  JOO;  what  was  the  nominal  va- 
lue of  the  notes  issued  in  these  three  months  ;  and  what 
was  the  cost  of  white  rags,  from  which  they  were  made, 
supposing  each  ounce  of  rag  might  be  manufactured  into 
twenty  five-pound  notes,  and  the  raij;s  to  be  worth  8c?. 
per  lb..?         Ans.  28,360.970  nominal,  590/.   17s.  0*^7. 

Ex.  47.  Two  persons  depart  from  London  to  York  on 
the  same  day  ;  the  one  walks  19  miles  a  day,  the  other 
only  15a  miles;  '^'^^  far  distant  will  they  be  from  one 
another  after  ten  days  travelling,  and  when  will  each 
get  to  York,  which  is  197  miles  from  London  ? 

Ans.  35  miles  distant,  Jie  who  goes  l9  miles  a  day 
will  complete  his  journey  on  the  IJth.  day,  whde 
the  other  will  not    complete  his  journey  till   the 
13th.  day. 
Ex,  48,   The  population  of  the  world  is  estimated  at  a 
thousand   millions  of  human  beings  ;  if  the  face  of  the 
earth  be  repeopled  every  33  years,  how   many   persons 
are  born  and  die  in  a  year,  week,  day,  and  minute  ? 
Ans.  30.303,030  year,  582,750^^  week,  832oOday. 
34118  nearly  in  an  hour,  58  nearly  a  nun. 
Ex.  49.  The   field  opposite  my  house  will  serve  50 
cows  forty  days  ;  how  long  will  it  afford  220  with  equal 
feed  ?  '  And.  9  days  and  a  fraction. 

Ex.  50.  If  10  persons  expend  250  dols.  in  4  months; 
how  much  ought  3  persons  to  expend  in  12  months  ? 

Ans.  225  dollars. 


(  1^9  ) 


FRACTIONS. 


A  Fraction  is  the  part,  or  parts  of  a  whole,  or  of  any 
wh.>l;*  quantity  expressed  by  unity,  and  is  expressed  by 
two  fifftjies,  with  a  line  drawn  between  them,  as  |,  J,  |. 

The  upper  fijjure  of  a  fraction  is  called  the  numerator, 
and  the  under  one  the  denominator. 

The  denominator  sliews  how  many  parts  the  unit  is 
divided  i  ito,  and  the  numerator,  how  many  of  these 
parts  are  to  be  taken  :  thus  |,  or  three-fourths,  shews 
that  the  wliole  is  divided  into  four  parts,  and  that  three 
of  those  i>arts  are  to  be  taken  :  and  f ,  or  five-ei«:hths, 
shew  that  the  whole  is  divided  into  eh^ht  parts,  and  that 
five  of  these  parts  are  taken. 

There  are  four  sorts  of  fractions,  simple  and  com- 
poun<i,  pr(»per  and  improper. 

A  simple  fraction  has  only  one  numerator  and  denomi- 
nator, as^,  or|. 

A  compound  fraction  consists  of  two  or  more  parts, 
and  is  known  '>v  the  word  of  placed  between  them,  as  | 
of  6:  or  I  of  |  of  ^%, 

A  proper  fraction  is,  when  the  numerator  is  less  than 
the  denominator. 

An  improper  fraction  is,  when  the  numerator  is  equal 
to,  or  greater  than,  the  denominator. 

A  mixed  number  is  formed  from  an  integer  and  a  frac- 
tion joined  together,  as  8^. 

A  complex  fraction  is  one  that  has  a  fraction  or  a  mix- 
ed number  tor  Us  nuaieratur)  or  deuomioator^  or  both. 


(  no  ) 
REDUCTION  OF  FRACTIONS. 


The  method  of  diangincr  fractions  from  one  form  to 
another,  without  altering:  their  value,  is  called  Reduc- 
tion ',^ZZIIii:[qZI:^ZZ.I.  Reduction  serves  to  prepare  frac- 
tions for  Addition,  Subtraction,  Multiplication,  and  Di- 
vision. 

Cask  1.  To  reduce  fractions  to  their  least  terms. 

Rule.  Divide  the  terms  of  the  given  fraction  by  any 
Dumber,  which  will  divide  them  both  without  a  remain- 
der, and  the  quotients  will  be  the  terms  of  a  new  frac- 
tion, equal  in  value  to  the  given  fraction.  Repeat  the 
operation,  till  the  terms  of  the  reduced  fraction  are  di- 
visible only  by  1.     ^ 

Ex.  1.  Reduce  |ii2  to  its  lowest  terms. 

)3136     392             \392     49  \49       7 
IT ,  and  8  ] — — ,  and  7  j — zr— 
3584     448              /448     56              J  5Q        8 
Keduce  the  following  fractions  to  their  lowest  terms. 
Kx.  1        Ex,  2.         Ex.  3.       Ex.  4.       Ex.  5. 
32        4    208      52      136      17    156      If-     S60     30 

120     15  684      171       72      9    356     28     708     59 

Ex.  6.       Ex.  7.  Ex.  8.        Fx.  h.         Ex.   10. 

384      1    5184     432  24-5     55  4032     224  3105        69 

1152     3     6012     ,01   2880     64 '4806     267  15705      267 
2X3X4X5 

Reduc? — to  its  lowest  terms. 

3X4X7X8 
2X3X4X5     -10      5 

3  X  4  X  7  X  8    65     28  -o 


DEDUCTION    OF    FRACTIONS.  141 

3X8X9X2 

Reduce — to  the  lowest  terms, 

4  X  3  X  14  X  36 
3X8X9X2      3x2x4x9x2   2   1 


4X3X14X30  4X3X2X7X4X9  28   14 
3X4X15X41 

Ex.  U. ZZ- 

5X6X24X3     2 
10  X  27  X  30  X  12     24 

Ex.  12. — — 

15  X    9  X  55  X  30     55 
Case  TI.  To  find  the  greatest  common  measure  of  a 
fraction. 

Rule.  Divide  the  greater  term  by  the  less,  and  this 
divisor  by  the  remainder,  then  the  fast  divisor  will  be 
the  greatest  common  measure  of  both  terms  of  the  frac- 
tion. 

■■4X,.    <w  i.u,.  i3  tiie  great«ak  %,„.,m^t%tiM  a*.wM0M^w  ^»  «... 

.fraction  i^^  ? 

918)1998(2 

1836 

162)918(5 
810 

108)162(1 
108 

\    91S      }7 

Ans.  54  greatest  C.  M.     54)108(2      51.  ) — — 

108  /iyy«    37 

What  are  the  greatest  common  measures  of  the  follow- 
ing fractions? 

Ex.  1.  Ex.2.  Ex.  3. 

270  1080  720 

Ans.  18     Ans.  72      Ans.  8 

306  1224  1736 

Ex.  4.  Ex.  5.  Ex,  6. 

3108  9600  14J60 

Ans.  444 \ns.  2400 Ans.  80 

3552         16800  16320 


142  REDUCTION    OF    FRACTIONS. 

Case  III.  To  reiluce  an  improper  fraction  to  an  equiva- 
«  lent,  whole,  or  mixed  number. 

Rule.  Divide  the  numerator  by  the  denomina^tor, 
and  the  quotient  will  be  the  integer,  or  mixed  number 
required  :  thus  'Jf=43,  and  ^=5. 

Reduce  the  following  improper  fractions  to  their  pro- 
per terms. 

Ex.1.     Ex.2.      Ex.3.      Ex  4.      Ex.5.      Ex.6. 
29  51  69  75  96  101 

--.1=3^.  —.=85.    — .-=8».    —.=6*.    —.=6. =:7ll 

8  7  8  12  16  13 

Ex,  7.  Ex.8.  Ex.9.  Ex.10. 

850  97()4.  5640  889 

.=35",     .  =  17/19. =12^. =2961 

24  556  450  3 

Case  IV.  To  reduce  a  mixed  number  to  an  equivalent 
improper  fraction. 

Rule.  Multiply  the   whole  number  by  the  denomina- 
tor of  the  fraction,  to  the  product  add  The  numerator,  for 
a  new  numerator,  under  which  place  the  denominator  ; 
Thus  4^=^,  and  29b|=«^». 

Reduce  the  following  mixed  numbers  to  their  equivalent 
improper  fractions. 

29  C9  77 

Ex.  1.  3^=—.     Ex.  2.  8^=—.     Ex.  3.  6/a=--. 

8  8  12 

101  .169                             6971 

Ex.  4.  7^3= Ex.  5.  1 8^,= Ex.  6.  43515— ■ 

13  9                  ^             16 

1135  2496 

Ex.  r.  378^= Ex.  8.  499:^= 

3  5 

1784  3853 

Ex.  9.  543^3= Ex.  10.  67|^= 

S3  57 


REDUCTION    OF    FRACTIONS.  143 

Case  V,  To  reduce  a  compound  fraction  to  an  equiva- 
lent simple  one. 

Rule  (1.)  If  any  of  the  proposed  quantities  be  inte- 
gers, or  mixed  numbers,  reduce  them  to  their  proper 
terms. 

(2  )  ^Multiply  all  the  numerators  together  for  a  new 
nucnerator,  and  all  the  deiiominatoi*s  lor  a  new  denomi- 
nator, and  then  reduce  the  fi  action  to  its  lowe^it  terms. 
Reduce  *  of  3  of  7^  to  a  simple  fraction. 

4  3         47        2  X  2  X  3  X  47      94 
Operation  —  x  —  X  —  = =  — . 

5  1  6         5X1X2X3  5 

The  fraction  ^is  already  in  its  lovvesst  terms,  because 
no  figure  higher  ilian  the  unit  will  divide  both  terms  of 
the  traction  without  a  remainder. 

Ex.  1.  I  off  of  5  of  1=1     Ex.  2. 1  of  4  of  32=.12f 

Ex.  3.  1^1  of  8  of  7^of  12=33l-iV 

Ex.  4.  ^  of  !s  of  1 2  of  9«=6A'^7. 

Ex.  5.  /e  of  10  of  »«^  of  18^=122*.  Ex.  6.  |  of  ^  of  ^of  ,«o=i.  1 

Case  VI.  To  reduce  fractions  of  different  denomina- 
tors to  others  of  equal  value,  having  a  common  denomi- 
nator. 

Rule.  (1.)  Multiply  each  numerator  into  all  the  de- 
nominators, except  its  own,^for  a   new  nuinerator,  and 
all  the  denominators  for  a  common  denominator. 
Reduce  %  b,  3l,  and  3,  to  a  corumon  denominator. 
Operation,  %  |,  \S  ^ 

New  nu- 
merators. 
3X9X3X1=     81 
7  X  5  X  3  X  1  =  J03 
11   X  5  X  9  X  I  =  495 
3X5X9X3=  405 

New  denom. 

5x9x3x1  =  rrs 

AUbwerj   ia5>  laaj  issj  isa* 


144  nenuoTioN  OF  fractions; 

Ex.  1.  Reduce  I,  ^.  and  l  to  a  common  denominator. 

A,,e      33      m      88 

2.  Reduce  J,  g,  *,  and  I,  to  a  common  denominator. 

Anau/or      2160     1400    2016    2305 

y\ijswci ,   2«zo^  agaoi  aeaot  sfiso* 

3.  Reduce  7,  g,  7,  and  3,  to  a  common  denominuior. 

Anewpr      7"    "7    ^lO    738 
miJ)Wer,     24S?  245*  246* 

4.  Reduce^,  io»  8,  and  llg.  to  a  common  denomina- 

fnv  A  neu'Ar     **     '">    2400   s4fio 

Lui.  rviidwri)   300*300'   300«   300' 

5.  Reduce  11,  2,  7,  4,  and  2^,  to;i  common  denomina- 

•*Or.  A  neurpr     ^^°    ^^    220    3080    16M 

lui.  Aribwer,  770'  770-  770    770*  77o» 

6.  Reduce  I,  %  7,  and  ^,  to  a  common  denominsitor. 

Ancu'Pr    *'»0    *8s    80     310 
/\nhuer»  gso^  23o»  sso?  280» 

7.  Reduce  L  9>  «?  «  and  7,  to  a  con^mon  denor-  i.'-i  or. 

AncwPr      1260     800       360       576      10080 
^IJHWtrr,     1410'   1410»   1440.    1440>    1+40  * 


660»  660* 

9.  Reduce  e,  i,  7,  and  *j,  of  9,  to  a  common  dinonnna- 

inr  Anuwpr       B^^      1388     623      l?098 

lor.  Answer,  sggo,  gegsj  3696>  sege* 

^^      (2.)  To  find  the  least  common  denominator. 

Set  down  the  denominators  ot  the  given  fraction?  in  a 
line,  and  divide  as  ninny  of  them  as  possible,  by  any 
number  v\bich  will  leave  no  remainder,  and  set  down  the 
quotients,  and  the  undivided  numbers  below.  Repeat 
the  operation  till  there  be  no  two  numbers  which  can  be 
divided  without  a  remainder.  Then  the  product  of  all 
the  divisors,  and  the  quotients  in  the  last  lines  will  give 
the  least  common  denominator.  Divide  this  least  com- 
mon denominator  by  each  of  the  given  denominators  se- 
parately, and  nmltiply  the  quotients  by  their  several  nu- 
mnators,  their  products  will  be  the  new  numerators. 
Reduce  %  I,  ",|,  to  the  least  common  denominator. 
3y5.  9,  3.  1 

,  then  3X5X3X1X1=45,  is  the  common 

5,  3,  1,  I 
non  denominator,  and  45  divided  by  the  given  denomi- 
nators, 5,  9,  3,  1,  give  9,  5,  15,  45;    these  multiplied 
by  the  given  nunierators,  give  27,  35,  165,  135,  for  new 
numerators,  and  the  fractions  will  stanU  S»  H,  ^^»  *^« 


REDUCTION    OF  FRACTIONS.  1^8 

Reduce  I,  1,  «  ti  and  1,  to  the  least  common  denomi- 
nator. 

3)3,  4,  5,  6,  8  The  least  denominator  is,  accordingly, 

—————  3  X4'X2x5==  120  • 

4)1,  4,  5,  2,  8  120-J-3,  4,  5,  6,  8=40,' 30,  24,  20,  15 

40X^5  30X3;  24x2;  20X4;   15x3, 

2)1,1,5,2,2  for    new  numerators;    therefore    the 

fractions  required  are  ^o  Z,  ^y  Z^  i^. 

1,1.5,1,1 

^  Case  VII.  To  reduce  a  fraction  of  one  denomination  to 
the  fraction  of  another  denomination  of  equal  value. 

Rule.  ( 1 .)  When  it  is  from  the  less  to  a  greater  deno- 
mination, '•  Multiply  the  denominator  by  all  the  deno- 
minations from  that  given  to  the  one  sought." 

Thus,  to  reduce  4  of  a  penny  to  a  fraction  of  a  pound, 
3  3 

the  answer  will  be  — -=  - — . 

4  X  12  X  20       960 

(2.)  When  it  is  from  a  greater  to  a  less  denomination, 
''  Multiply  the  numerator  by  all  the  denominations,  from 
that  given  to  the  one  sought.'* 

Thus,  to  reduce  ^  of  a  pound  to  the  fraction  of  a  farthing, 
6X20X12X4     5760 


7  7 

Ex.  1.  Reduce  ^  of  a  farthing  to  the  fraction  of  a 
pound.  Answer,  8^. 

2.  Reduce  9  of  a  penny  to  the  fraction  of  a  shilling. 

Answer,  i^* 

3.  Reduce  g  of  a  pound  to  the  fraction  of  a  farthing. 

Answer,  ^. 

4.  Reduce  ^  of  a  pound  to  the  fraction  of  a  penny. 

Answer,  ^. 

5.  Reduce  l\  of  a  pound  to  the  fraction  of  a  farthing. 

Answer,  *T- 
6*  Reduce  3  shillings  to  the  fraction  of  a  pound. 

Answer,  ^ 
13 


140  REDUCTION    OF    FRACTIONS. 

7.  Reduce  9  of  a  dwt.  to  the  fraction  of  a  lb.  Troy; 

Answer,  g^o* 

8.  Reduce  5  of  a  cwt.  to  the  fraction  of  an  ounce. 

Answer,  1280  oz.=80lb. 

9.  Reduce  ^  of  a  week  to  the  fraction  of  an  hour. 

Answer,  "J^ 

10.  Reduce  4  of  a  mile  to  the  fraction  of  a  yard. 

Answer,  1320  yards. 

1 1 .  Reduce  s  of  a  pipe  to  the  fraction  of  a  gallon. 

Answer,  ^*. 

12.  Reduce  *  cent  to  the  fraction  of  a  dollar 

Ans.  200  dollar. 

Caes  VIII.  To  find  the  value  of  a  fraction  in  numbers  of 
inferior  denomination. 

Rule.  Multiply  the  integer,  or  its  value  in  the  next 
lower  denomination,  by  the  numerator,  and  divide  by 
the  denominator  : 

3  X  20 

Thus,  the  value  of  i  of  a  pound  is  equal  to =12 

£  X  12       5 
shillings,  and  §  of  a  shilling  equal  to =  8  pence. 

•      3  ^*" 

Ex.  1 .  What  is  the  value  of  9  of  a  pound  ?  Ans.  llsAld. . 

2.  What  is  the  value  of  ^  of  a  shilling  ?     Ans.  lojc?. 

3.  What  is  the  value  of  ^  of  half  a  crown  ? 

Ans.  18c?, 
4..  What  is  the  value  of  ^  of  a  lb.  Troy  ? 

Ans.  9  ounces. 

5.  What  is  the  value  of  ^^  of  a  cwt.  ?    Ans.  72  lb. 

6.  What  is  the  value  of  9  of  a  mile  ? 

Ans.  977^  yards. 

7.  What  is  the  value  of  f  of  a  cwt.  ?  Ans.  48  lbs. 

8.  What  is  the  value  of  ^  of  a  dollar?     Ans.  4l|cts. 

9.  What  is  the  value  of  7  of  a  hogshead  of  wine  ? 

Ans.  54  gallons. 

Case  IX.  To  reduce  a  complex  fraction  to  an  equivalent 
simple  fraction. 


REDUCTION    OF    FRACTIONS.  .  H7 

Rule.  If  the  numerator  or  denominator,  or  both,  be 
whole  or  mixed  numbers,  reduce  them  to  improper  frac- 
tions ;  and  multiply  the  denominator  of  the  lower  frac- 
tion into  the  numerator  of  the  upper,  for  a  new  numera- 
tor, and  the  denominator,  of  the  upper  fraction  into  the 
numerator  of  the  lower  for  a  new  denominator.    ^ 

4  *         4X8       52  *o       ^0         4 

'    I  I         7X1        1  5       I  50 

.    51      'i      4r.  9       ?      G3  51 

And — = — = — .  And  — ~ — = — .  And  again — 51— 

8        ?       64  ■  3^      f       23  3^ 

!^       147 

— = .    No.  other  varieties  can  occur. 

'^       96 

31 
Ex.  i.    Reduce  —  to  a  simple  fraction.  Ans.  ie 

4 

3 
4 

2.  Reduce  —  to  simple  fraction.  Ans.  ^^. 


2,  Reduce to  a  simple  fraction.        An.  ^. 

1%         ^ 

4.  Reduce to  a  simple  fraction.       Ans.  ^. 

DO 


Ex.  5.  Reduce  —  to  a  simple  fraction.  Ans.  j^ 

6.  Reduce  —  to  a  simple  fraction.        Ans.  i^i 

5 

7.  Reduce  —  to  a  simple  fraction.  Ans,  ^l 

7 


148 


ADDITION    OF    FRACTIONS. 


4 
8.  Reduce to  a  simple  fraction.       Ans.  i^ 

•  19^ 


ADDITION  OF  FRACTIONS. 

Rule.  Reduce  mixed  numbers  to  improper  fractions , 
and  compound  or  complex  fractions  to  simple  ones,  and 
bring  them  all  to  a  common  denominator.  Add  all  the 
numerators  together,  and  write  the  sum  over  the  com- 
mon denominator. 

Ex.  Add  I,  3,  5*,  and  I  together  ;  which  is  thus  per- 
formed :  I  I,  ",  I 

3X3X2X4=     72' 

2X5X2X4=     80/     Therefore  ^^  +  ^  +  Ifo 


■I 


]  I  X  5  X  3  .X  4  =  6601  -^  Z  ==  f^l  =  7,^  =  7^, 
1X5X3X2=     30  r  which  is  the  answer. 

5X3X2X4=  120  ; 
This  may  be  performed  by  bringing  the  given  fractions 
to  the  least  common  denominator. 
2)5,  3,  2,  4, 

Thus,  %  I,  ")  «  then ,  and  the  new  deno- 

5,  3,  1,  2, 
minator  zi  60  ;  the  fractions  will  beiS  +  62+M+w=? 

60     =    «60' 

Ex.  1.  Add^,  ?,  and  ;  together.  Ans.  l=r« 

S.  Add  t,  I,  and  I  together.  Ans.  5^^^ 

S.  What  is  the  sum  of  i,  *,  and  4>l?      Ans.  5% 

4.  Add  together  3^,  41,  and  I.  Ans.  8',^ 

5.  Add  ^,  «,  21  and  5l  together.  Ans.  9^ 

6.  What  is  the  sum  of  7i,  3*,  and  |.     Ans.   1 1|', 

7.  What  is  the  sum  of  7  of  a  guinea,  g  of  a  shiU 
ling,  and  g  of  a  penny  ?  Ans.  0/.  9s.  5Jrf. 

8.  What  is  the  sum^  of  g  of  a  pound,  7  of  a  shilling, 
and  ^  of  a  penny  ?  Ans.  0/.  5s ^  Q^d.  |i 


I 


SUBTRACTION    OF     FRACTIONS.  149 

Ex.  9.  What  is  the  sum  of  4  of  a  guinea,  3  of  a  shil- 
ling, and  10  <jf  a  penny  ?  Ans.  16s.  diod. 

10.  If  I  have  8  of  a  coasting  vessel,  and  purchase 
another  share  of  11,  what  part  of  her  will  belong  to  me  ? 

Ans.  ^ 

11.  Add  3  of  a  yard,  and  4  of  a  mile  together. 
Ans.  1320  yds.  2  feet. 

12.  What  is  the  sum  of  3  of  a  yard,  |  of  a  foot  and 
I  of  an  inch  ?  Ans.  2  ft.  9  in.  2  b.c. 

13.  Add  eof  a  lb-  troy  to  I  of  an  ounce. 

Ans.  2  oz.  15  dwts.  ]2grs. 

14.  What  is  the  sum  of  |  of  an  eagle  and  i  of  a 
dollar  ?  Ans.  8  dols.   10  «ts. 

15.  Add  8  of  dollar  to  ^  of  a  cent?  Ans.  56^i  cts. 


.oflONS. 
SUBTRACTION  OF  ^ 

-ce  the  given  fractions  to  the  same  deno- 
RuLE-as  In  Addition,  then  subtract  the  lesser  nume- 
."ffi'or  from  the  greater,  and  under  the  difference  place 
the  common  denominator.    ^  „  ^        ,. 

Kx.  Take  I  from  ,\  :  and  ^  from  H 

5  X    9*^  45—36      9         I 

3  X  12  Wherefore == =—  Answer. 

—  V  108        108      12 

9  X  123 


11    X    15") 
9     X  I6f 


165—144      21      7  ' 

Therefore  • ^=— -=— , 

240        240     80 


15   X   16) 
Ex.  1.  From  I  take!.  Ans. 

2.  From  r^  take  ^  Ans. 

3.  From  ii  take  L.  Ans.- 

13** 


1 


350  MULTIPLICATIOK    OF    FRACTIONS. 


3<i' 


4.  From  *|  take*.  Ans-   ^ 

5.  From  9U'ike  47s.  ^ns.  4|. 
•6.  From  1^2^  take  I  of  17.  Ans.  1„.^ 

7.  From  ^  of  asltilinj^  take  ^^,  of  a  pound.  Ans.3/i. 

8.  From  I  of  a  pound  take  l^  of  a  pound.     Ans.  ^. 

9.  From  I  take  /a-  Ans.  as. 

10.  From  1  take  ^  of  ^  Ans.  15. 

11.  From  12  take  ^.  Ans.  11 9. 

12.  From  10/.  take  |  of  a  pound.  Ans.  9i.  8s.  lO^jcf. 

1 3.  From  |  of  a  pound  take  i^  of  a  pound. 

Ans.  6s.  6a. 

14.  From  %  of  a  pound  take  ^  of  ^  of  a  shilling;. 

Ans.  13«.  Q\d. 

15.  From  %  of  6  dollars  take  ^of  5  dollars. 

Ans.  2  dols: 

Subtract  ^%  cts,  from  8$  dollars. 

Ans.  8  dols.  lie  cts. 


^*^^'r^^^^^-^-  "  OF  FRACTIONS. 

Rule.  Reduce  mixed  nunnbers  to  imp.  .^ 
and  compound   fractions  to  simple  ones  ;    mfri^ftions, 
the  numerators  to^^elher  for  a  new  numerator ;  and  aL. 
the  denominators  for  a  common  denominator. 

Ex.  Multiply  3%  %  and  I  of  8  together. 

29        3        .5        8         29  X  3  X  5.  X  8       29  X  5       145 


X  —  X  — X  — ~ 


8  4         6         1  8X4X3X2        4X2  8 

-=  ISg,  the  answer. 

Kx,   1.  Multiply  M  by  9 ;  and  %  by  f^,    Ans.  ^^  and  ^. 

2.  What  is  the  product  of  ^  %  and  3^  ?     Ans.  3i|. 

3.  What  is  the  product  of  57  by  ^^  .^     Ans.  46/i. 

4.  What  is  the  product  of  71  multiplied  by  33  .^ 

Ans.  259 

5.  What  is  the  product  of  ^,  f,  121,  and  ^  of  10  ? 

Ans.  irj. 


DIVISION    OF    FRACTIONS.  13 1 

6.  What  is  the  continued  product  of  ^.  ^,   5J,  and 
<^?  Ans.  16  ^. 

7.  What  is  the  product  of  ^  of  ^,  ^  of  It  ? 

An».  ^*,. 

8.  What  is  the  product  of  ^,  1,  tt «  L  9  and  »,  ?  ' 

Ans;  ^. 

9.  How   many  yards  are  there    in  54  pieces    of 
Irish,  ench  cont^ininw  2t>\  ?  Ans.  ]3^\l. 

10.  How  m:ny  pounds   are    there  in  83  cheeses, 
each  containinor  £5^  lb,  Ans.  2 18^. 


DIVISION  OF  FRACTIONS. 

RuLv.  Reduce  the  fractions,  as  in  MuUiplication  ^ 
tht >n  ID  ert  the  divisor,  and  proceed  as  in  Multiplication  : 
thus,  I  to  be  divided  by  9. 

3     .      3  3w9  27   9 

^"^   »  - —  ft    ^   3   \B    fi« 

Ex.   Divide^  of  4,^hy^of  1. 
3        23         3         1      3  XS.'i  3* 

8        5         7        48X5       4X7 
=s>^;;,'t=  i6io  the  answer. 

EX.VMPLES. 

tx.   I.    Divide  If  of  12  by  I.  Ans.  7lf, 

2.  Divide  A  of  8  by  ^.  Ans.  22^. 

3.  Diviile  f  by  ^,J.  Ans.   5^. 

4.  Divide  I  of  64  by  J.  Ans.  i^iO. 
f-  ^ivide|of  12byS.?.  Ans.   jf?. 

^":':;^l«f  36by3i.  Ans.  8^. 

Ans.  fg. 


7.  Divide  ^^  4bvlof  2. 
M|i^i^%»'|b^lof^  Ans 

9.  Divide  ^  of  ^  of  5bv?of  L 

10.  Divide /,  of  ^  by  ,Vof  5. 

11.  What  nu .Tiber  multiplied  by  f  will  give  9*  ? 


Ans.  15 
An&.   «^7: 


16* 
1100* 


Ans. 


2.  What  number  multiplied  by  ^  of  3  will  give  56  ? 

An^  44|» 


152 


PRACTICE. 


13.  What  number  multiplied  by  ^  of  f  of  15  will 
produce!  of  4?  Ans.  i^g. 

1 4.  From  5  subtract  ^  of  f  of  4|  and  divide  the  re- 
mainder by  4.  Ans.   l^^. 

13.  What  is  a  person's  share  of  a  prize  of 
jL.2O,0uO  gths  of  which  is  to  be  divided  among;  13  per- 
sons? Ans.  1230/.  15s.~4|rf.  ^^ 


PRxiCTICE. 

Practick  is  a  method  of  finding  the  value  of  any 
quantity  of  goods,  from  the  p'rice  of  an  integer  being 
given. 

ALiquoT  PARTS  of  any  number  or  quantity,  are  such 
as  will  exactly'  divide  it  without  leaving  a  remainder: 
thus  7  and  4  are  aliquot  parts  of  28,  4  pence  is  an  ali- 
quot part  of  a  shilling,  and  5  shillings  is  an  aliquot  part 
of  a  pound. 


TABLES  OF  ALIQUOT  PARTS. 


IMOt 

parts  of  a  L. 

Parts  of  a  shil. 

Parts  of  3  pence. 

S. 

d. 

d. 

q- 

10 

0  ==i 

6     =     i 

8         1 

4         '4 

6 

8     =3^ 

4     =     I 

1  r  °. 

5 
4 
3 

2 

0  =  I 
0  =  1 

4  =  S 

6  =  i 

3     =.     1 

2     =     I 

1         =r      13 

4     —      la 
,  Parts 
of  a  penny. 

\  =  t 

0  =/o 

Farts 

4        *=        4.- 

8  =i 

of  a  sixpence. 

4  -.*» 

*  =  1 

3  =  ?« 

i  =  i 

0  =i 

PRACTICE. 


163 


I.  When  the  price  is  less  than  a  penny. 

Rule.     Divide  the  quantity  hy  the  aliquot  parts  in  a 
penny,  and  the  quotient  by  12  and  20.    ^ 

Ex.  What  is  the  value  of  7853  yards  of  tape,  at  I  per 
yard  ? 

7853 


392')* 
1963^ 


12)58891 


2,0)49.0     91 


Answer,  X.24  10  9^ 


■                                           EXAMPLES 

• 

L,    s. 

d. 

Ex7l.456rat^peryd. 

Answe 

r,  4  15 

11 

2.  6784  at  i  per  lb. 

14    2 

3.  3976  at  i 

12     8 

6 

-     4   7655  at  ^  per  yd. 

15   18 

m 

^5.  7486  at  1  per  lb. 

S3     7 

101 

6.  9984  at  1 

20  16 

0 

7.  6327  atiper  yd. 

19  15 

51 

8.  5934  at  i  per  lb..' 

6     3 

n 

9.  7585  att 

./ 

15  16 

01 

10.  4767  at  1  per  yd. 

/ 

4    19 

31 

11.  6493  at  1  per  lb. 

20     5 

91 

12.  5388  at  i 

16   16 

9 

II.  When  the  price  is  an  aliquot  part  of  a  shilling. 

Rule.     Divide  the  s:iven  number  hy  the  aliquot  part, 
aad  tliis  quotient  by  20  :  the  answer  will  be  in  pounds. 


154 


PRACTICE. 


Ex.  What  is  the  value  of  2785  lbs.  of  salt  at  4i.  perjb.?- 


W. 


2785 


2.0)92.8  4 


Answer,  LA6  8  4 


Ex. 


i.     s. 

d. 

1.   3764at2(£. 

Answer,  31     7 

4 

2.  5943  at  3d, 

74     5 

9 

3.  4953  at  V^d. 

30  19 

n 

4.  5943  at  4d. 

'  99,    1 

0 

5.  3987  at  3d, 

49   16 

9 

6.  5964  at  \d. 

24  17 

0 

7,  5684  at  Ad, 

94  14 

8 

8.  2105  ^tQd, 

22   10 

10 

9.   3456  at  2i. 

28  16 

0 

10.  3924  at  I|(i. 

37     0 

6 

11.  5904  at  2(/.          ^ 

49   14 

0 

12.  5215  at4f/. 

86  18 

4 

JII.     "When  the  price  is  pence  and  farthings,  and  no 
aliquot  part  of  a  shilling. 

RuLK.  (1.)  Find  what  aliquot  part  of  a  shilling  is 
nearest  to  the  given  price,  and  divide  the  proposed  num- 
ber by  it.  (2)  Consider  what  part  the  remainder  is  of 
this  aliquot  part  of  the  given  price,  and  divide  the  for- 
mer quotient  by  it,  &c.  (3)  Add  the  several  quotients 
together,  and  the  answer  will  be  in  shillings,  which  di- 
vide by  20  to  bring  into  pounds. 

Ex.  What  is  the  value  of  4277  yds.,  at  i02^.  per  yd.?- 


6 

.1 

4277 

3 

1. 

2138 

6 

34     |\l069 

5 

i 

i       534 

7i 

89 

H 

2,0^ 

38:vl 

5| 

Answer, 

L. 

lyi  11 

^i 

^^^■V                          FRACl 

:icE. 

1 

d. 

L. 

s.     d. 

Ex.  1.  4784  at    U     , 

Answer,  24 

18  '4 

.  2.  5964  at    \% 

43 

9     9 

3.  4659  at    2* 

43 

13     62 

4.   J765at    2\ 

16 

10    ill 

5.  4305  at    21 

49 

6     62 

6.  3694  at    35 

50 

0     Si 

7.  7641  at    2^ 

79 

11    \ol 

8.  9875  at    6^ 

267 

8   11* 

9.  5476  at  10^ 

245 

5     7 

10.  3592  at    3^ 

52 

7     8 

11.  3046  at    62 

85 

13     4* 

12.  32l4at  111 

154 

0     1 

13.  8764  at    3^ 

136 

18     9 

14.  5921  at    1\ 

178 

17     3* 

15.  5178  at    91 

204 

19     3 

16.  9714  at    41 

182 

2     9 

17.  5643  at    8^ 

199 

17      1* 

18.  4932  at  10* 

210 

12     9 

19.  8934  at    5\ 

195 

8     71 

20.  2458  at    9i 

99 

17     1* 

21.  8764  at  11| 

429 

1      5 

22.  5687  at    51 

136 

5     Oi 

23.  1435  at  10* 

62 

15     71 

24.  5842  at    7i 

176 

9     6* 

25.  5943  at   9| 

235 

4   10* 

26.   187  6  at    2| 

21 

9    11 

27.  43 16  at    7^ 

139 

7     5 

28.  1956  at    82 

71 

6     3 

29.  4235  at    5\ 

97 

1     01 

30.   1327  at    9^ 

52 

10     6* 

31.  2748  at  11 

125 

19     0 

32.  9374  at    7 J 

283 

3     5* 

33.  4285  at  ll^ 

200 

17     2* 

34.  1594  at   3* 

23 

4  11 

35.  5632  at  5' 

117 

6     8 

36.  U14at  5* 

25 

10     ? 

155 


156 


PRACTICE. 


^  '    IV.  When  the  price  is  more  than  one  shilling,  and  Iqss 
than  two. 

Rule.    Let  the  given  number  stand  f©r  shillings,  and 
work  for  the  pence  and  farthings  as  bt?fore.     ^ _, 

Ex.   What  is  the  value  of  1187  quartern  loaves,  at  Is' 
lid,  each  ? 


Ex 


H 

i 

11187 

i 

I 

6" 

1     J48     4^ 
24    al 

2 

,0)136.0  \l 

Answer  L.68  0  l\ 

s. 

d. 

L.    s.    d. 

'.   1.  3456  at  1 

21 

Answer,  208  1 6    0 

2.  487«>  at  1 

51 

355   10  10 

3.  5792  at  1 

8^ 

494  14     8 

4.  2632  at  1 

35 

172  14     6 

5.  4092  at  1 

71 

328     4     3 

6.  2.-96  at  1 

]0 

237   19     4 

7.  4735  at  I 

n 

325  10     7| 

8.  3724  at  I 

91 

333   12     2 

9.  3451  at  1 

61 

269   12     21 

10.  7321  at  I 

72 

602     9     12 

11.  5928  at  1 

11 

568     2     0 

12.  6542  at  1 

81 

S6r,   12     21 

13.  8465  at  1 

9^ 

758     6     5^ 

14.  4371   at   1 

31 

282     5   10^ 

15.  8937  at  I 

SI 

586     9     92 

16.  1234  at  1 

11 

lis     5     2 

17.  5629  at  I 

n 

322     9   102 

18.  4516  at  1 

2 

263     8     8 

19.  5678  at  1 

2^ 

348  19     21 

20.  9272  at  1 

42 

647    0     9  J 

21.  5461  at  1 

7 

432     6     7 

2S.  8234  at  1 

51 

600     7   11 

23.  5i)28  at  1 

lOl 

555  15     0 

24,  8750  at 

1 

5 

619  15  10 

r 


PEACTIOE.  1^7 

When  the  price  is  any  number  of  shillings  under  20. 

Rule.  (1.)  If  the  price  is  an  even  number,  multiply 
the  given  quantity  by  half  the  said  number,  doubling  the 
first  figure  to  the  right  hand  for  shilling^;,  and  the  rest  are 
pounds.  (2.)  If  the  price  is  an  odd  number,  find  for 
the  greatest  even  number,  as  before,  to  which  add  the  j^th 
•:,  of  the  given  number  for  the  odd  shUling,  and  the  sum  is 
the  answer.      / 

Ex.  What  is  the  value  of  3456  yards  of  cloth,  at  18s, 
per  yard  ? 

34.56 
9 


Ans.  L.  3110  8 

Ex.  What  is  the  value  of  2592  yards  of  second  cloth? 
at  lis.  per  yard? 

l  =  ^\  2592 
5 


1296     0 
129  12 


Answer  L.  1425  12 

EXAMPLES. 

Ex.  I.  S'^rs  at    2  Answer 


s. 

I. 

6'^75  at 

2 

2. 

4374  at 

3 

3. 

5916  at 

4 

4. 

7691  at 

5 

5. 

6743  at 

6 

6. 

9430  at 

8 

7. 

6734  at 

10 

8. 

594f)  at 

11 

9. 

3004  at 

7 

10. 

2yc^5  at 

13 

11. 

4392  at 

14 

12. 

5931  at 

19 

L. 

S. 

597 

10 

6J6 

2 

1183 

4 

1897 

15 

2022 

J8 

3772 

0 

2867 

0 

3270 

6 

1051 

8 

1907 

15 

3074 

8 

5634 


14 


PRACTICE. 

s. 

L. 

s. 

13.  491  r  at  18 

Answer 

44.25 

6 

14.  3271  at     9 

1471 

19 

15-  9515  at  17 

7917 

15 

16    2514  at    16 

£011 

4 

17.  1392  at  10 

696 

0 

18.  54S2  at  19 

5160 

8 

VI.  AVhen  the  price  is  shillings  and  pence. 

Rule.  (1.)  If  they  are  an  aliquot  part  of  a  pound, 
divide  the  quantity  by  that  part,  and  the  quotient  is  the 
answer.  (2.)  If  they  are  not  an  aliquot  part,  multiply 
by  the  shillings,  and  take  parts  for  the  pence. 

Ex.  What  is  the  value  of  2769  yards  of  Irish,  at  3s. 
4d.  per  yard  ? 

3s.  4d.  I  2769 


Answer  L.  461  10s. 

Ex.  "What  is  the  value  of  3758  yards  of  muslin^  at 
I2s«  9d,  per  yard  ? 


6 
3 

1 1      3756 
12 

*       4-072 

1878 

J         938 

Answer 

2.0)47889 

L.  2394    9s. 

EXAMPLES. 

%. 

S 

1.  8943  at    S 

2,  3532  at     4 
S.  «67l  at    7 

4.  2524  at    3 

5.  5971  at     5 

6.  5460  at     7 

.    d. 

5    0          Answer 

0 

6 

91 
10 

6 

L. 

894. 

706 
3251 

478 
1741 
2047 

s. 
6 
8 
12 
10 
10 
10 

d. 
0 
0 
6 

10 
0 

^^B' 

PRACTICE. 

s» 

d. 

L.      s. 

d. 

Ex.  7.  S764  at  10 

0 

Answer  1  882  0 

0 

8.  5638  at  8 

n 

2513  12 

2 

9.  S7+5  at  9^ 

11 

1856  17 

U 

10.  8756  at  15 

10 

6931  16 

8 

11.  31)42  at  4 

5 

870  10 

6 

12.  2475  at  16 

8 

2062  10 

0 

16.  5642  at  18 

4i 

5183  11 

9 

14.  1764  at  5 

8 

499  16 

0 

15.  5931  at  17 

6 

5189  12 

6 

19.  9J43  at  6 

8 

304-7  13 

4 

17.  7  189  at  3 

7 

1288  0 

7 

18.  4004  at  19 

6 

4488  18 

0 

159 


VII.  When  the  price  is  pounds  and  shillings,  or  pounds; 
shillini^s,  pence,  and  farthings. 

Rule.     Multiply  the  quantity  by   the  pounds,  and 
work  the  rest  by  the  foregoing  rules. 

Ex.  What  is  the  value  of  5428  hogsheads  of  ale,  at  4>l, 
12s,  per  hogshead  ? 

5438 
4   12 


21712 
3256  16 


Answer,  L.  24968  16 

Ex.  What  is  the  value  of  2714cwt.  of  sugar,  at  5U 
12s.  v9^.  per  cwt.  ? 


10s. 

1 

2 

2714 
S 

8142 

25.  6d. 

1 
4 

1357 

3d, 

1 
.10 

339  5  0 

t 
a 

1 

33  18  6 

5  13  1 

Answer,  L,  9877  16    7 


/ 

leo 

PRAGTICE. 

L.    s. 

d. 

L. 

s. 

d'. 

Ex.  1.  5674  at 

5   17 

6 

33334 

15 

2.  6431  at 

4  8 

4 

28403 

11 

8 

3.  3416  at 

5  11 

6^ 

19U54 

17 

6 

4.  4^31  at 

9  4 

0 

45365 

4 

5.  S146  at 

10  12 

9 

334'>5 

11 

6 

6.  4316  at 

10  19 

Gl 

4737  7 

1 

10 

7.  5648  at 

12  13 

0 

71447 

4 

8.  1436  at 

10  10 

6 

15113 

18 

9.  1346  at 

3  13 

4 

4935 

6 

8 

10.  2714  at 

18  9 

0 

500."  3 

6 

11.  9614  at 

4  14 

6 

45426 

3 

12»  5789  at 

7  7 

7 

42717 

19 

11 

13.  1590  at 

12  12 

0 

20034 

14.  6341  at 

8  18 

6 

56593 

8 

6 

15.  4S03  at 

9  9 

91 

45583 

9 

5i 

16.  3465  at 

8  15 

0 

50.3 18 

15 

17.  7<8£  at 

11  12 

10 

83610 

9 

0 

18.  1604  at 

4  11 

10 

7S65 

0 

8 

VIII.  If  there  be  a  fraction  in  the  given  quantity* 
Rule.    Work  for  the  whole  number,  according  to  the 

preceding  rules,  to  which  add  1,2,1,85  &c.  of  the  price, 

according  to  the  nature  of  the  question. 
Ex.  .What  is  the  value  of  5354|  cwt.  of  soap,  at  U.  4^. 

Sd.  per  cwt.  ? 


4s. 


Bd. 


5354  I 
4 

21416 
1070  16 
178  9 
3  3 

4 
6 

4  4  8 

2  2  4 
1  2  2 


L.3  3  6 


Answer,  L.22668     8  10 


Ex. 


L.  s,  d.                L.  s.  d, 

1.  4562i  at  3  15  9^  Ans.  17289  0  6l— 3 

2.  6744^  at  9  9  lU^  64030  11  II* 

3.  26541  at  7  15  4  20618  11  2 

4.  73941  at  12  8  SJ  91949  1  10* 

5.  465  U  at  5  12  10  26240  16  0^ 
§.  3749^  at  16  9  5  61757  7  9^ 


PHAeTICE. 


161 


Z.  s,    d. 

Ex.    7.  3875    at     8   18     a^ 
8.  43()5l  at    ll    11    11 
9    9724i  at     6   16     4l 
10.  36t8i  at     4     4.     6^ 


L.     ,  s. 

Ans.  34596  9 
50624.  10 
66307  4 
15426  6 


7 

3^ 


Aliquot  part  of      Aliquot  parts 

a  ton. 

ot  a 

cwt. 

cwt.  qr.  lb. 

qrs 

.  lb. 

10    0    ozz  I 

2 

0-  I 

5     0     0=1 

1 

0  -  I 

4    0    0  z:  I 

10  -  ,* 

2     3   12  —  ^ 

14-^ 

2     2     0  —  1 

8- A 

2    0    0  ::=  /o 

7-^ 

1    0    0— i 

Aliquot  parts 
of  a  lb. 
oz. 


TABLES  OF  ALIQUOT  PARTS. 

Aliquot  parts 
of  a  qr.  of  cwt. 
lb. 

14    -1 
7    —1 

*  =7^ 
ol  1 

•>9   8 

*     14 

13    1 

*4  —  16 

1     -^ 

IX.  When  the  given  quantity  is  of  several  denominations* 

Rule.  Multiply  the  ^iven  price  by  the  highest  deno- 
mination, as  in  Compound  Multiplication,  and  take  parts 
of  the  price  for  the  inferior  denominations  of  the  given 
quantity. 

Kx.  What  is  the  value  ^23  cwt.  3  qr.  21  lb.  of  hops^ 
at  4/.  \8s.  Qd.  per  cwt.  ? 


2  qr. 


1  qr. 

I4lb  . 
71b. 


l\ 


18 


d. 
6 
11 


54     3 


Here,  for  the  22  cwt,  I  multi- 
ply by  1 1  and  by  2  ;  then  1  take 
parts  lor  the  3  qi-s.  21  lb.,  accord- 
ing to  the  preceding  table,  and 
by  case  Vlll. 


108     7 

0 

ZZ  value  of  22  cwt 

^  2     9 

3 

:ZI  ditto       2  qrs. 

1      4 

n 

ZZ  ditto         1  qr. 

12 

Si 

Z:  ditto      14  lb. 

6 

\i- 

irz  ditto       7  lb. 

iUis.  L.  U:^  19    4  —  1 


14* 


162  PRACTICE, 

cw4:.  qrs.  lb.     dolls,  cts. 
Ex,    I.     8     '2  14    at  20  50        per  cwt. 

Ans.  176  ('oils.  Slfcfs. 

2.  16     1   21     at   14  80         per  cvt. 

Answ«r,  243  «Jo11s  27^  cts. 

3.  37     3  22     at  12  ll     7  per  cut. 

Answer,  L.  477     6     8. 

4.  73     2  10^   at     3  16     9  per  cut. 

Answer,  L,  282     8     S], 

5.  38     1    16     at     2  12     6  per  cwt. 

Answer,  L   100  15     7^. 

6.  ^i     2     8     at  39     3     8  per  cwt. 

Anf^wer.  L.  1315     8     % 

7.  84     3   14     at  12   11     8  por  cwt. 

Aoiwer,  L.  106S     0     2j. 
/>.     s.    d. 
6.     56  tons,    4  cwt.    S  qrs.    Olb.  at58     7    6  per  ton. 

Answer,  L.  3282     2     8^, 
9.     39tons,  12  c^S^t.    1  qr.    14lb.  at23   12    8  per  ton. 

Answer,  L.  10J5   1 1     2. 

10.  124  tons,  16  cwt.    2  qr.    16lb.  ail2   18     7  per  tun. 

Answer,  L.  1613   19     6  nearlv, 

11.  16  lb.        8  oz,    12tlr.    -    -     at    4     3     6  per  lb. 

Answer,  L.  m     1     7\, 

12.  25  lb,      12  oz.       4  dr.    -     -      st    8   12     6  per  lb. 

AHlswer,  L.  222     4     6i. 
IS.     35  lb.        4  oz,    12dwt.-     -     at  1 1     9     9  per  lb. 

Answer,  L.  4.06     9     3^ 

14.  48  lb.        8 '>z.    16dwt,-     -     at  14     4     4  per  lb. 

Answer;,/.,  692   16     5|. 

15.  25  lb.        6oz.      5dwt.  -     -     at  15     3     9  per  lb. 

Answer,  L.  387   I J    Hi, 

16.  18  yds.     2qr.      Snails-     -     at    0  16     b  per  yd. 

Answer,  L,  \5   11     5*. 

17.  55yds.     3  qr.     Snails-     -     at    1     3     9  r#    yd. 

Answer,  L.  66     7     Oi. 

18.  l5acF.      3rd.  24  per.  -    -      at  38     3     6|jeraiT« 

Answer,  L.  606   19     7^ 
§,     25  acr,       1  rd.     4  per.  -     -     at  22  50    0  per  acr. 

Answer,  568  dulU,  68|    ts. 

20,    59  acr      3rd.  18  per.  •     -      at  36  25    0  per  acr. 

Answer,  1317  Uuii&.  lliaCtS* 


(  1G3 


TARE  AND  TRET. 


Tare  and  Tret  are  a  set  of  practical  rules  for  de- 
ducting; c»'rtuin  allowances,  made  by  wholesale  dealers 
in  selling  their  goods  hy  weight. 

Guo'^s  Weight  is  the  whole  wei2;ht  of  goods,  includ- 
ing package,  or  whatever  contains  them. 

Neat  Weight  is  what  remains  after  all  allowances 
are  made. 

Take  is  an  allowance  to  th#buyer,  for  the  weight  of 
the  package,  and  is  either  at  so  much  per  barrel,  chest, 
&c  ,  or  at  so  mucii  per  cwt.,  or  at  so  much  for  the 
whole. 

Tret  is  an  allowance  of  41b.  in  every  i04lb.  for 
waste,  vlust,  &c.,  or  the  i^  part  of  the  whole. 

Cloff  is  an  allowanci,  after  Tare  and  Tret  are  de- 
ducted, of21h.  upon  every  3  cwt.  that  the  weiglit  may 
hold  good   when  sold  by  tiie  retail. 

Suttle  is  when  only  part  of  the  allowance  is  deduct- 
ed from  the  gross.  I'hus,  after  the  tare  is  deducted 
from  the  gross,  the  remainder  is  calhd  tare  suttle. 

Case  1.  When  the  tar-e  is  so  much  for  the  whole. 

KuLE.  From  the  gross  weight  subtract  the  tare,  and 
ihe  reinaiuder  will  be  tiie  neat  weight  required.     . 


164  TARE    AND    TRET. 

Ex.  What  is  the  neat  wei^'it  of  25  barrels  af  in(Ti^(iy 
weiuihin;^  116  cvvt.  2  qr.  14  lb.,  allowing  2  cvvt.  3  qr. 
12  lb.  tare  ? 

Gwt.  qr.  lb. 

IJ6    2     14 

2    3     12 


Answer  -  113  3       2  neat  weio;ht. 

Fx.  1.  What  is  the  neat  weit;ht  of  55  barrels  of  figs^ 
weighing  35  cwt.  2  qr.  !J  lb.,  tare  beiny;  allowed  at 
I  cwt.   I  qr.  24  lb.?  Ans.  34  cwt.  0  qr.    19  lb. 

Ex.  2.  What  is  the  neat  vvei;^ht  of  20  casks  of  Ilussiair 
tallow,  weighing  74  cwt,  tare  be*ng  allowed  at  2  cvvt. 
2qr.  3  lb.?  Ans-  71  cwt.   1  qr.  23  lb. 

Case  II.    When  the  tare  is  at  so  much   per   barrel, 
chest,  &c. 

Rule.  (\.)  Multiply  the  tare  by  the  number  of  hogs- 
heads*' barrels,  chests,  &,c.  subtract  the  product  from 
the  )rvnss,  and  the  remalider  will  be  the  neat  weight  re- 
quired :  or 

(2.)  Subtract  the  tare  of  each  parcel  fron^.  the  given 
weight,  and  multiply  by  the  number  of  parcels. 

Ex.  What  is  the  neat  weight  of  8  hhds.  of  tobacco, 
each  weighing  4  cwt.  2  qr.  24  lb.  gross,  tare  being 
allowed  at  2  qrs.  4  lb.  per  hhd.  ? 

cvvt.  qr.  lb.  qr.  lb. 

4.     2    24.  2     4 

8  8 


Gross  weight     37     2     24  4  1    4  Tare. 

4     1       4 


a 


Answer    -    33     1    20  rieat  weight* 

Ex.  1.  What  is  the  neat  weight  of  tS^frails  of  Mala- 
ga raisins,  each  weij>hing  2  cwt.  3  qrs.  12  lb.,  when  the 
Ure  upon  each  trail  is  17  ib.?  Aua.  07  cwt.  2  qr.  15  ib« 


TARE     AND    TRET. 


16^ 


Ex.  In  79  barrels  of  fi2;s,  each  weiojhinj;  ]  cwt.  12  lb. 
and  tare  91b.  per  barref,  what  is  the  neat  weight  ? 

Ans.  81  cwt.  C  qr.   13  lb.  neat  weight* 

Ex.  3.  What  is  the  neat  weight  of  24  hhtls.  of  tobac- 
co, the  weight  of  each  being  4.^  cwt.,  and  tare  67  lb.  per 
hhtU?  Ans.  93  cwt.  2  qr.   iO  lb.  neat  weight. 

Ex.  4,  In  18  casks  of  currants,  each  weighing  6  cwt. 
1  qr.  12  lb.  and  tare  61  lb  per  cask,  what  is  the  neat 
weight  ?  Ans.  104  cwt.  2  qr.  14  lb.  neat  weight. 

Case.  III.  When  the  tare  is  at  so  much  per  cwt. 

Rule.  Take  the  aliquot  part  or  parts  of  the  whole 
gross  weight  that  the  tare  is  of  a  cwt ,  as  in  Practice,  and 
subtract  the  result  from  the  gross  weight. 

Ex.  What  is  the  neat  weight  of  24  barrels  of  figs,  each 
weighing  3  cwt.  2  qrs,   12  lb.  and  tare  12  lb.  per  cwt.  B 

cwt.  qr.  lb. 

X  24  =  6 


3     2 


Tare, 


Ans  77 


21 

2 

16 
4 

.86 

2 

8 

oz. 

,   9 

1  . 

2 

13^ 

X  4 

lb. 

8 


cwt. 
86 


qr. 

2 


,  lb. 

8 
—  oz. 

20    9 
10    4i 


9    1     2 


5       2\  nt.  wt. 


13i 

[tare. 


Ex.  1.  What  is  the  neat  weight  of  21  barrels  of  pot- 
ash, each  barrel  weighinu;  1  cwt.  3  qr.  8  lb.,  tare  being 
10 lb.  per  cwt.?  Ans.  34  cwt.  3  qr.  9^  lb.  neat  wt, 

Ex.  2.  What  is  the  neat  vveight  of  35  barrels  of  ancho- 
vies, each  weighing  1  qr.  12  lb.,  tare  at  14  lb.  per  cwt.? 
Ans.    10  cwt.  3  qr.   21  lb.  neat  weight. 

Ex.  3.  Requrttd  the  neat  weiu,ht  of  15  hhds.  of  tobac- 
co, each  weighing  4  cwt.  2  qrs.  12  lb,  tare  at  20  lb. 
per  cwt.  ?    Ans.  56  cwt.  3  qr.  2  lb.  neat  weiglit  nearly. 


166 


TARE    AND    TRET. 


Ex.  Jn^What  is  the  value  of  26  hogsheads  of  tobacco,^ 
at  8/.  5|s.  per  cwt.  each  hogshead  weighing  4^1  cwt.^ 
and  the  allowance  for  tare  being  J  3  lb.  per  cwt,  ? 

Ans,  8 68/.   14.S.  6d, 

Case  IV.  Wlien  there  is  an  allowance  both  of  tare»and 
tret. 

HuLE.  Find  the  tare  by  the  last  rule,  subtract  it  from 
the  i;n)ss  weight,  the  remainder  or  suttle,  divided  by  26, 
gives  the  tret,  which  being  subtracted  from  the  suttle^ 
gives  the  answer.  ^ 

Ex.  What  is  the  neat  weight  of  15  casks  of  tallow, 
each  weighing  0  cwt.  2  qr.  12  lb.,  tare  being  12  lb.  per 
cwt.  and  tret  as  usual  ? 

cwt.  qr.  lb. 
6       2     12  X  15  =  5  X  3. 
5 


33 

0 

4 
3 

-6 

lb.        owl*  qr.  lb. 
8  1  A     99     0     12 

4  1  ^       7     0     8  — 12 
1           3     2     4—    6 

Gross  wt.  99 
Tare  -     10 

0 
2 

12 
12 

3 

2 

1 

0 
17- 

10     2    12—18 

Answer    85     0     10  neat  weight. 

Ex.  I.  In  18  cwt.  1  qr.  6  lb.  gross,  tare  63  lb.,  and 
tret  as  usual,  how  much  neat  ? 

Ans.  17  cwt.  Oqr.  7  lb.  neat  weight. 

Ex.  2.  In  14  casks  of  raisins,  each  2  cwt.  14  lb.  gross, 
tare  18  lb.  per  cwt.,  and  tret  as  usuaL  what  is  the  neat 
weight  ?  Ans.  24  cwt.  0  qr.  1  lb.  neat  weight. 

Ex.  3.  In  9  chests  of  sugar,  each  weighing  8  cwt. 
2  qr.  10  lb.,  tare  14  lb.  per  cwt,  and  tret  as  usual, 
what  is  the  neat  weight  ?  Aus.  64  cwt.  3  qr  24  lb.  nt.  wt.- 


TAllE    AND    TRET.  167 

Case  V.  When  cloftis  allowed, 

Pule.  Subtract  the  tare  from  the  r,ross,^Rj(]  the 
tret  frOiii  the  taro  suttle  ;  then  liiviHc  the  tre."  suttle  by 
168,  and  the  result  will  he  the  iJlotf,  which  l>eirjir  sub- 
tracte'l  from  the  last  suttle,  gives  the  neat  weigiit  re- 
quired. 

Ex.  What  is  the  neat  weij;ht  of  19  cwt.  1  qr.  2  Jh. 
gross,  tare  3  cwt.  3  qr.  2-1  lb.,  and  tret  and  cloft*  at  the 
usual  rate  ? 

cwt.  qr.  lb.         cvvt.  qr.  lb. 

Gross  -19       12  4)14     2     26-T-168— 4x6x7 


Tare    -       3       3  22 


26)15        18 


6)3     -Z     20  8oz. 


Tret  -  2  10  7)2     12  12 


Tret  suttle  14        2  26  9  13^ 

Cloff  -  9   13  oz. 


Ans.  cwt.    14       2  16     3  neat  weight. 

Ex.  1.  What  is  the  neat  weight  of  224  cwt.  3  qr.  20lb. 
of  tobacco,  tare  beinji;  25  cwt.  3  qr.,  tret  and  cloff'  as 
usual.  Ans.   cwt.  190   I    14  neat  wei•^!lt. 

Ex.  2.  Inl4hhds.  of  tobacco,  each  weighing  5  cwt. 
3  qr.  17  lb.  gross,  tare  11  lb.  per  cwt.,  and  tret  and 
cloll*  as  usual,  what  is  th€f/nrtit  weij^ht  ? 

Ans.   cwt.  70  2  2  neat  weight. 

Ex.  3.  What  is  the  neat  weight  of  15  casks  of  cur- 
rants, each  weighing  Si  cwt.  gross,  tare  35  lb.  per  cask, 
tret  and  cloff  usual?  Ans.  cwt.  74  J    14  neat  weight. 

F.K.  4.  In  9  chests  of  sugar,  each  containing  7  cwU 
2qr.  12  lb.  gross,  tare  13  lb.  per  cwt..  tret  and  cloii  as 
usual,  what  is  the  neat  weight,  and  what  is  the  value  of 
it  at  9^.  per  lb.  ? 

Aus.  cwt  67  3  14  nt.  wt         256/.    lis.  7d. 


(  108  ) 


0 


DECIMAL  FRACTIONS. 


1.  Decimal,  or  Decimatfd  Fpaotions,  are  such  as 
always  h.ivt  I  cith  one  or  nsor^  (jf-he's  for  their  deuo- 
iT)i'fi.*^ors.  The  denoiti';r»ator«  are  n«ner  express«?(l.  being 
uinlerstood  to  b^  10,  100,1000,  ifec,  acrorditit:  a^  the 
nu!>iprators  consist  ot"  1,  </,  or  fi^urrs  :  thus,  ini^tea^i  of 
10'  iTO?  wxioy  the  numerators  only  are  wraren.  ^vith  a  dot  or 
\\v.   rtt>(J  comma  before  then>,  as  2:    24;    211. 

2.  Ha  (iecin»al  consists  of  only  t.ne  fimirts  one  is  sup- 
p.  -  d  o  be  divided  into  ten  equai  partsi  and  the  decimal 
rep!  -sents  as  many  ot  those  parts  as  fhe  decimal  fijiure 
expresses  ;  thus,  J  nu  ans  seventi-nths  of  an  unit :  If  it 
copsist  of  two  fi^;ures,  one  is  supposed  to  be  divided  into 
1XX>  equal  parts,  of  which  the  tlecir.tai  represents  as  manjr 
as  tl  e  figur**  expresses  :  thus,  .65  means  sixty-five  hun- 
dredths of  an  unit. 

3.  (Jypiiers  to  the  right-hand  of  decimals  cause  no 
diffierence  in  their  valut,  for  .5  ;  .50  ;  500,  are  decimals 
of  the  saa.e  vahie.  ben.g  each  equal  to  \ ;  that  is,  .5  —  io5 
.50  =  1^;  .500  =  ,^ ;  hut  if  Ae  cypliers  are  placed  on 
the  left-hand  of  aecid.als,  they  dinnnish  their  value  in  a 
ten-fold  proportion,  thus  .3  ;  .03  x  .003,  are  3-tenths,  3- 
hundredths  ;  S-rhousaudths  ;  and  answer  to  tht  vuigar 
fractious  I,,  i^,  i^o,  respectively. 

4.  A  whole  iiufi'l'er  and  decimal  is  thus  expressed^ 
85  74  which  is 

8.^>74  8504 

equal  to  85ioo  = and  45.04  =»  85^^  =  —      -  ^-c. 

100  100 


(  169  > 


REDUCTION  OF  DECIMALS. 


Case  I.  To  reduce  a  vulgar  fraction  to  Q^ecimal  of 
an  equal  value.  ''^^* 

Rule  Divide  the  numerator  of  the  fractrtwEi'increased 
by  a  cypher,  or  cyphers,  by  the  denominator,  and  the 
quotient  will  be  the  decimal  sought. 

Keduce  |,  ^,  ^^  jV»  to  decimals  of  the  same  value. 
1=10  =,.5.  1^1:00^.25.  1=^1.000^,125. 

tV=  ^?°-  =  .0625. 

The  cyphers  avlded  to  the  numerators  are  separated 
from  the  original  figures  by  a  dot,  to  shew  that  they  are 
borrowed  for  the  sake  of  forming  the  decimal. 

Ex.  1.  What  decimal  expre^^sioas  answer  to  the  fol- 
lowing vulgar  fractions,  |,  |,  |,  f ,  ||  ? 

Ans.  |-ooo  =  375.  f  ^°^  =  .623.  |--ooo  =  .875.  f-^o 
=  .222,  &c-  11-000  =  .733,  &c. 

Ex.  2.  Required  the  equivalent  decimals  of  the  frac- 

finnc       5  9        3       7  9 

Ans.  If  =  .2.  -I  0000  ^  .5625.  |-oo  =  .75.  -^'Oo* 
=  a363,&c.  T?^  =  1  =.5. 

Ex.  3.  What  is  the  decin\al  that  answers  to  -^^  ? 

Ans.     T>^  =  10^0000  ^  .015625. 

Ex.  4.  What  are  the  decimals  answering  to  the  frac- 
tions T-l^,  aVV;*  and  ^^^  ? 

An^      {■^■s:  =  .0390625.  |fg-  =  .05859375.  ^^|^ 
=  .019531,  &c. 

Ex.  5.  What  decimal  expressions  answer  to  ^,  -^ 
and  /^Jj  ? 

All!,.     -}  =  .333,  &c.  -^  =  .020202,  &c.  ^Vt- 
=  .123123123,  &c. 

Case  11.  To  reduce  numbers  of  different  denominations 
to  their  equivalent  decimal  values. 
Rule.   (1)   Write  the  given  numbers  under  each  other 
for  dividends,  proceeding  from  the  least  to  the  greatest 
15 


/: 


4 
12 
20 


179  REDUCTION    OF    DE«IMALS* 

(2)  Place  on  the  left  side  of  each  dividend,  for  a  divisor, 
the  number  that  will  bring  it  to  the  next  superior  deno- 
mination. (3)  Begin  with  the  uppermost  number,  and 
set  down  the  quotient  of  each  division,  as  decimal  parts, 
on  the  right  hand  of  the  dividend  next  below  it,  and  so 
proceed  to  the  last  quotient,  which  is  the  decimal  re- 
quired. 

Ex.  Reduce  12s.  3^d.  to  the  decimal  of  a  pound. 
3  qrs,  I  divide  the  |  by  4,  supplying  cy- 

Sd.,75  phers  to  the  3  by  the  imagination  ; 

12s..  3 125  the  quotient  is  .75,  which  is  placed 

_  by  the  side  of  the  3d.f  and  then  di- 

.615625  >         vide  the  3SS  by  12;    the  quotient, 
decimal  of  a  L.  J         .3125,  1  set  by  the  side  of  the  12s, 

«— and    divide    by    20,    which    ^ives 

.615625  for  the  answer  :  that  is,  if  a  pound  were  divided 
into  1,000,000  parts,  the  12s.  3|rf.  would  be  615625  such 
parts,  in  the  same  manner  as  if  a  penny  were  divided 
into  100  parts,  |  would  be  equal  to  75  such  parts. 

Ex.  1.  Reduce  8s.  4|d.  to  the  decimal  of  a  pound. 

Answer,  .41875. 

2.  What  decimal  of  a  pound  is  15s.  5|(i.  .♦^ 

Answer,  .77395833,  &c. 

3.  What  decimal  of  a  pound  is  4s.  6if/.  ? 

Answer,  .22604166,  &c. 

4.  Reduce  18s.  Od.,  8s.  2d,,  and  5s.  to  decimals  of 
a  pound. 

1st.  Ans.  .925.     2d.  Ans.  .40833,  &c.    3d.  Ans.  .25. 

5.  Reduce  5  oz.  6  dwts.  8  gr.  troy,  to  the  decimal 
of  a  pound.  Answer,  .443055. 

6.  Reduce  3  qrs.   7  lb.  8  oz.  avoirdupois,  to  the 
decimal  of  a  cwt.  Answer,  .816964. 

7.  Reduce  2  qrs.  1  n.  to  the  decimal  of  a  yard. 

Answer,  .o625. 

8.  Reduce  3  pks.  1  gal.  2  qts.  to  the  decimal  of  a 
bushel  ?  Answer,  .9375. 

Case  III.  To  find  the  value  of  any  given  decimal  in 
ierms  ©f  the  integer.  This  is  the  reverse  of  the  last 
case. 


ADDITION  OF  DECIMALS' 


171 


Rule.  Multiply  the  decimal  by  the  number  of  parts, 
in  the  next  less  denomination,  and  cut  off  as  many  places 
to  the  ri2;ht-hand,  as  there  are  places  in  the  given  deci* 
mal,  and  so  proceed  through  each  denomination. 

Ex.  What  is  the  value  of  .615625  of  a  pound  ? 
.615625 

It  may  be  observed,  that  as  cyphers  to 
the  right  do  not  alter  the  value  in  deci- 
mals, they  are  omitted  in  each  step  «f  the 
operation. 


20 


12.312500 
12 


V^- 


3.7500 
4 


3.00 
Ex. 


Answer,     12s.  3|d. 


1.  What  is  the  value  of  .625  of  a  shilling  ? 

Answer,  7^  penc'^ 

2.  What  is  the  value  of  .1275  of  a  pound  ? 

Answer,  2s.  6|d.  4. 

3.  What  is  the  value  of  .575  of  a  cwt. 

Answer,  2  qr.  8  lb.  6  oz.  6  dr 

4.  What  is  the  value  of  .875  of  a  bushel  ? 

Answer,  3  pks.  1  gal. 


4. 


ADDITION  OF  DECIMALS. 


Rule.  (1.)  Arrange  the  numbers  under  each  other, 
accordmg  to  their  several  values.  (2.)  Find  the  sum  as 
in  Addition  of  whole  numbers,  and  cut  off,  for  decimals) 
as  many  figures  to  the  right  as  there  are  decimals  in  any 
on^  of  the  given  numbers. 


\A 


472  SUBTRACTION  OF  DECIMALS. 

Ex.  What  is  the  sum  of  23.45,  7.849,  543.2,  8.6234 
and  253.004  ? 

23.45 
7.849 
543.2 

8.6234 
253  004 


Answer,  836.1264 

Ex.  1.  What  is  the  sum   of  37.035,  4.26,  598.034, 

9.3076,  4.321,  and  5  ?  AnswiM-,  (J57.9o76. 

2.  Find  the  value  of  39.33,  4.2056,  .987 35,  46.x:87, 

3,7491,  and  8.004.  Answer,  97.56305; 


SUBTRACTION  OF  DECIMALS. 


Rule.  Arrange  the  numbers  according  to  their  value  ; 
subtract  as  in  whole  numbers,  and  cut  off  for  decimals, 
a^  in  Addition. 

Ex.  Subtract  35.87043  from  132.005. 
132.005 
35.87043 


Answer,    96.13457 

Ex.  1.  What  is  the  difference  between  I04.3'?6  and  74.05  ? 

Answer,  30  ^276 
Ex.  2  Find  the  difference  between  394.832  and  14H.0076. 

Answer,  246.8244 
Ex.  3.  From  3r2.97 1  take  270.30041 . 

Answer,  102.6705'9 


(  173  ) 


MULTIPLICATION  OF  DECIMALS. 


Rule.  Multiply  as  in  whole  numbers,  and  cut  off  as 
many  figures  from  the  product  as  there  are  decimals  in 
the  multiplier  and  multiplicand, 

Ex.  Multiply  .025  by  .045  :  also  4.82  by  3.53. 

'Ji                „t„  In  the  first  instance,  there 

• _[ being  but  four  figures  in  the 

-g^              ~  .^.g  product,  and  six  decimals  in 

*  10  ^-'^^    multiplier  and  multipli- 

\±Afi  cand,  two  cyphers  must  be 

*  ***^  added  to  the  left  hand  of  the 


100  2.410 


.001125 


17.0146       P'^^"^^- 


Ex.  1.  Multiply  76.43  bv. 875:  also  .897  by  .452. 

Answers,  66.87625— -.405444 
Ex.  2.  Multiply  324.004  by  .7  872 

Answer,  255.0559488 
Ex.  3.  What  is  the  product  of  9.57  and  .074  ? 

Answer,  .70818 
Ex.  4.  Multiply  .643  Ijy  .389 

Answer,  .250127 

When  the  number  of  decimals  in  the  multiplicand  is 
large,  and  it  is  not  wished  to  carry  the  operation  to  more 
than  a  certain  number  of  decimals  in  the  product,  it  is 
done  by  the  following  Rule,  which  I  shall  illustrate  by  an 
example. 

Rule.  Having  arranged  the  multiplicand,  count  as 
many  figures  from  the  decimal  point,  as  you  intend  to 
keepdecirnals  in  the  product,  and  make. a  *  over  the  last  of 
these,  under  which,  after  you  have  inverted  the  multi- 
plier, place  the  units  figure  of  the  multiplier  thus  invert-?|^ 
ed,  and  the  others  in  their  proper  order.  Then  multiply  V^;- 
each  figure  of  the  inverted  multiplier,  beginning,  as  usual 

M 


174 


MULTIPLICATION  OF  DECIMALS. 


at  the  right  hanH  and  set  down  the  respective  products  s© 
that  the  right  hand  figures  may  fall  in  a  straight  line  under 
one  another.  In  multiplying,  no  attention  is  to  be  paid 
to  the  figures  on  the  right  hand  of  that  which  you  multi- 
ply by,  unless  it  be  with  the  two  preceding  figures,  td 
find  what  number  should  he  carried. 

Ex.  Required  the  product  of  1.570796,  multiplied  by 
26.3719,  with  four  places  of  decimals  in  the  product. 
This,  in  the  usual  method,  would  yield  ten  places  of  de- 
cimals ;  by  contraction  it  is  thus  performed. 


1.570796 
9.17562 


314159  31  product  with  2  regard  being  had  to  2  x  6 

94247  rz 6  — ' 6X9 

4712ZI 3 3x7 

1(,99  :zz 7 


41.4246 


263719 


We  will  now  work  the  example  in  the  common  way. 
1.570796 

From  this  it  will  appear  plain, 
why  in  the  contracted  form  the 
multiplier  is  inverted  :  the  last 
product  here  being  the  first  there. 
In  the  contracted  form,  the  units 
place  is  6  ;  it  would  however  be 
8,  if  the  2  were  carried  from  the 
27,  obtained  in  the  next  line  bj 
Addition. 

Ex,  2.  Multiply  128.678,  by  33.24  sous  to  have  but  one 
|}lace  ^f  decimals. 


14 

137164 

15 

ro796 

1099 

557« 

4712 

388 

94247 

§ 

314159 

2 

41.4248 

75  >324 

DIVISION   OF    DECIMALS. 

Common  method.        Contracted  method. 


17S 


128.678 
38.24 

128  678 
42.83 

514712 
25^306 
1029421. 
386034 

38603 

10294 

257 

51 

4920.64672 

4920.5 

DIVISION  OF  DECIMALS.* 


Rule.  (1  )  Divide,  as  in  whole  numbers,  and  cut  off 
as  many  figures  in  the  quotient,  as  the  decimal  places  in 
tiie  dividend  exceed  those  of  the  divisor.  (2.)  If  there 
be  notfij^ures  enouj^hin  the  quotient,  the  deficiency  must 
be  supplied  by  prefixing  cyphers,  (3.)  If  there  be  a  re- 
mainder, or  there  i)e  more  decim  il  places  in  the  divisor 
than  in  the  dividend,  cyphers  may  be  affixed  to  the  divi- 
dend, and  the  quotient  carried  on  to  any  extent. 
Divide  1.71 54  Hy  1.5  ;  and  .37046  by  16. 
1.5)1.7134         16)  i7046  In  the  first  example,  by 

■  —        _-^— -  supplying  a  single  cypher 

1.1436  .02315375  there  is  no  remainder  left; 

but  in  the  second  I  must 
sunplv  three  cyphers  to  obtain  an  even  answer ;  and  I 
find  the  quotient  has  one  figure  less  than  there  are  deci- 
mals in  the  dividend  *(»  supplied.  I  must  therefore  prefix 
a  cypher  to  the  quotient  found. 

KO'lE. 

*  The  Contracted  method  of  Divison  maybe  thus  performed. 

Rule.  Having  determined  how  many  places  of  whole  ly 
here  will  be  ^n  the  quotient,  if  any,  which  is  easily  know 
section  ;  if  there  are  nooe^  then  consider  of  what  v^ 


176 
Ex. 


DIVISION    OF    DECIMALS. 


Ans.  7.0343  nearly. 

Ans.  4936.7ii||. 

Ans.  5.64i77  nearly. 

Ans.  .01992,  &c. 


1.  Divide  25.64  by  3.645. 

2.  Divide  4732  by  .9587. 

3.  Divide  .865439  by  .156. 

4.  Divide  79  by  3965. 

5.  Divide  3.3.64472  by  882.  Ans.  .038146,  nearly, 

6.  Divide  .218  by  7.435.  Ans.  .0293,  &c. 

7.  Divide  76.42  by  58.  Ans.  1.317,  &C' 

8.  Divide  88  by  .88.  Ans.  lOO. 


first  figure  in  the  quotient  will  be,  and  proceed  as  in  common 
Division,  only  omitting  one  fij^ure  of  the  divisor  at  each  opera- 
tion ;  viz.  for  every  figure  of  the  quotient  dot  off  one  in  the 
divisor,  remembering  to  carry  for  the  increase  of  the  figures  cut 
off,  as  was  done  in  Multiplication. 

Ex.  Let  it  be  required  to  divide  23.41  by  7.9863. 
Contracted  method.  i'ommon  method. 
7.9863)23  4100(^2.9312  |  7.9863)23  410U(2  9312 
15.9726  15.9726 


Here  it  must  .74374 
be  observed,  that  71876 

in    each  of  the 

subtractions  ex-  .2497 
cept  the  first,  .2395 
unit  mustbe  car-  -— 
ried  to  the  first  .101 
figure,  as  would  79 
be  the  case  in  — — 
the  usual  course.  .21 
15 

w5- 


.74374 
71876 


.2497,30 
239   189 


.101 
79 

.21 
15 


410 

865 

.470 
t'726 


.5  5744 


(  17/) 

REDUCTION  OF  DECIMALS, 


To  chang;e  the  currencies  of  the  different  states  to 
"Federal  money,  and  Federal  money  to  currency  by  de- 
cimals. 

1. — To  reduce  Maryland,  Pennsylvania,  Delaware, 
and  New  Jersey  currencies  to  Federal  Money. 

*  Rule.  Reduce  the  given  sum  to  the  decimal  of  a 
pound,  and  divide  by  .375  the  quotient  will  be  the 
answer. 

EXAMPLES. 

Fx.  1.  Reduce  76i.  Hs.  6d*  Maryland  currency  to 
Federal  money  ? 

12)6 


2,0)14.5 


.375)76.725(204.6  or  204  dols.  60  cts. 
75  0 


1725 

1500 

— \ 

2250 
22j0 


Ex.  2.  Reduce  237/.  17s.  4i.  Pennsylvania  currency 
to  Federal  money  ?  Ans.  dols.  634.3111  &c. 

Kx.  3.  Reduce  673^  Is.  2d,  New  Jersey  currency 
to  Federal  money  .?  Ans.  dols.  1794)8222  &c. 

Ex.  4.  Reduce  7/.  6s,  8rf.  New  Jersey  currency  to 
Federal  money  ?  Ans.  dols.   19.5555  &c. 

*  NOTE.  As  7s  6rf.  of  this  currency  make  a  dollar,  reduce 
it  to  the  decimal  of  a  pound,  and  it  will  be  .375/.  the  divisor 
ghren  in  this  ruler. 


178  HEDUCTION    OF    DECIMALS. 

2-— To  change  Federal  money  to  Maryland,  Penn- 
sylvania, Delaware,  and  New  Jersey  currencies. 

Rule.     Multiply  the  given  sum  by  .375  and  the  pro- 
duct will  be  pounds,  which  reduce  to  shillings  and  pence. 

EXAMPLES. 

Ex.  1.  How  much  Maryland  currency  in  S76.50  ? 
76.50 
•375 


38250 
53550 
22950 


L.    28.68750 
20 


13.75000 
12 


9.00000 
Ans.  28i.  13s.  9d, 

Ex.  2.  Change  744  dols.  into  Pennsylvania  currency  ^ 

Ans.  279Z. 
3.  Change  365.25  dols.  into  Pennsylvania  cur- 
rency ?  Ans.  136^  19s.  ^d. 
Ex.  4.  Change  627.75  dols.  into  Mairvland  currency  ? 

Ans.'^235^  8s.  \\d. 


8. — To  change  New  England  and  Virginia  currencies 
to  Federal  money. 

*  Rule.  Reduce  the  given  sum  to  the  decimal  of  a 
pound,  and  divide  by  .3,  the  quotient  will  be  the  answer. 


*  As  6  shillings  of  this  currency  make  one  dollar,  reduce 
6  shillings  to  the  decimal  of  a  pound,  and  it  will  be  .3,  the  di- 
visor given  in  the  rule. 


KEDUCTION   OF    DECIMALS.  179 


EXAMPLES. 

Ex.  I.  imU,  68.  8rf.  New  England  currency,  how 
much  Federal  money  ? 

12)8 

2,0)6.6' 
,3)74.333' &c. 

g  247.77'  &c. 

Ex.  2.  In  64Z.  J  5S'  Virginia  currency,'  how  much  Fe- 
deral money  ?  Ans.  dols.   215.833  &c. 

Ex.  3.  In  3271.  l6s.  ^d.  Virginia  currency,  how  much 
Federal  money  ?  Ans.  dols.   1092.722  &c. 

Ex.  4.  In  463^  12s.  9d.  Virginia  currency,  how 
much  Federal  money  .^       Ans.  dols.  1545.45833  &c. 

4 To  change  Federal  money  to  New  England  and 

Virginia  currencies. 

Rule.  Multiply  the  given  sum  by  .3,  and  the  pro- 
duct will  be  pounds,  which  reduce  to  shillings  and  pence. 

examples. 

Ex.  1.  Change  273.35  dols.  to  New  England  currency  ? 
273.25 
.3 


b  1.975 
20 

19,500 
12 


6,000      Ans.  81/.  19s.  6d. 
Fix.  2.  Change  496  dols.  to  New  England  currency? 

Ans.  148/.  16s, 
3.  Change  79.50  dols.  to  Virginia  currency  .? 

Ans.  2.d.  17 s, 

4k  Change  673.60  dols.  to  Virginia  currency  ? 

Ans.  202/.   Is.  7.2rf. 


180  REDUCTION   OF    DECIMALS. 

5.  To  change  New- York  and  North-Carolina  curren- 
cies to  Federal  money. 

*  Rule.  Reduce  the  given  sum  to  the  decimal  of  a 
pound,  and  divide  by  .4-,  the  quotient  will  be  the  answer. 

Examples. 

Ex.  1.  In  74L  l6s.  New- York  currency  how  much  Fe^ 
deral  money  ? 

2,0)  1 6 

•  .4)74.8 

gl87  Ans. 

Ex.  2.  In  29^  17s.  New-York  currency,  how  much 
Federal  money  ?  Ans.  g74.625 

Ex.  3.  In  365/.  7s.  4d.  New-York  currency,  how 
much  Federal  money?  Ans.  g9l3.4l()6.  &c. 

Ex.  4  4y7/.  16s.  \0d.  North  Carolina  currencv,  how 
much  Federal  money  ?  Ans.  gl244  604166,  &c. 


6. — To  change  Federal  money  to  New-York  and  North 
Carolina  currencies. 

Rule.  Multiply  the  given  sum  by  .4  and  the  product 
will  be  pounds,  the  decimal  parts  of  which  reduce  to 
shillings  and  pence. 

Examples. 

Ex.  1.  Reduce  S49  50  to  New-York  currency  ? 
49.50 
.4 


19.800 
20 


I6.U00  Ans.  19?.  165. 


♦  Note.  As  8  shillings  of  this  currency  make  one  dollar,  re- 
duce 8  shillings  to  the  decimal  of  a  pound,  and  it  will  give  .4 
the  divisor  given  in  the  rule. 


REDUCTION    OF    DECIMALS.'  181 

Ex.  2.  Reduce  g246  to  North-Carolina  currency  ? 

Ans.  98/.  8s. 
Ex.  S.  Reduce  S418.75  to  New-York  currency  ? 

Ans.  167/.  105. 
Ex.  Reduce  &847.60  to  New-York  currency  ? 

Ans,  339/.  09,  9.6d. 


7. — To  change  South-Carolina  and  Georgia  currencies 
to  Federal  money. 

*  Rule.  Reduce  the  given  sum  to  the  decimal  of  a 
pound,  multiply  by  30  and  divide  the  product  by  7  ;  the 
quotient  will  oe  the  answer. 

Examples. 

Ex.  1.  In  69/.  I5s.6d,  South-Carolina  currency,  how 
much  Federal  money  ? 

12)6 
2,0^15.5 


69.775 
30 


)  2093.250 


299.03571428' 

Ex.  2.  In  864/.  l7s.  2d.  South  Carolina  currency,  how 
much  Federal  money  ?  , 

Ans.  23706.535714285'. 

•  Note. — As  reducing  the  currency  of  these  states  to 
the  decimal  of  a  pound,  would  produce  a  circulating  de- 
cimal, I  have  formed  this  rule  on  the  principle  of  Vul- 
gar Fractions. 

4s.  Srizz^V  or  1^  of  a  pound,  consequently  the 
proportion  will  stand  thus  -^qL  :  \doll. ::  pounds  :  dollars, 
or  as  7  is  to  30  so  are  pounds  to  dollars,  agreeably  to  tht 
rule. 

16 


182  REDUCTION    OF    DECIMALS. 

Ex.  3.  In  9271.  16s.  9d.  Georgia  currency,  how  much 
Federal  money  ?  ^        , 

Ans.  S3976. 4.4642857 1 

Ex.  4.  In  6.73^.  12s.  8d.  Georgia  currency,  how  much 
Federal  money  ?  , 

Ans.  2886.99  or  2887  dollars. 

8. — To  change  Federal  money  to  South-Carolina  and 
Georgia  currencies. 

Rule.  Multiply  the  given  sum  by  7,  and  divide  that 
product  by  30,  the  quotient  will  be  the  answer  in  pounds ; 
the  decimal  parts  of  which  reduce  to  shillings  and  pence. 

Examples. 

Ex.  1.  How  much  Georgia  currency  in  %2i6.50  ? 
g216.50 
7 


3,0)151,5.50 


50.516'  circulates 
20 


10.33'  &c. 

12 
3.99'  &c,    Ans.  50/.   lOs.  4<^. 

Ex.  2.  How  much  South-Carolina  currency 
in  8467.25  ?  Ans.   109^.  Os.  6d. 

Example  3.  How  much  South-Carolina  currency 
ing762.S?  Ans.  177/.  17s.  4.8rf. 

Example  4.  How  much  Georgia  currency 
ing939.7?  Ans.  219/.  5s.  3.19'd: 

9.  To  change  Canada  and  Nova«Scotia  currencies  to 
Federal  molfiey. 


REDUCTION    OF    DECIMALS.  183 

*  Rule.  Reduce  the  given   sum  to  the  decimal  of  a 
pound;  and  divide  by  .25  the  quotient  will  be  the  answer  ? 

Examples. 

Ex.  1.  Reduce  STL   16s.  4rf.  Canada  currency  to  Fe- 
deral money  .^ 

12)4. 

2,0)1,6.3 


.26)87. 8l6'(S351.26'&c. 
75 

128  or  thus, 
125  {.5)87.816' 
.25 


(.5)1' 


31  t,5)n56i 

25 


g35l.26'  &c. 


66 
50 

166 
150 


16 

Ex.  2.  Reduce  827^  15s.  Nova-Scotia  currency  to 
Federal  money  .^  Ans.  g33ll. 

Ex.  3.  Reduce  268^  12s,  Sd.  Canada  currency  to  Fe- 
deral money  .^  g  1074.45. 

Ex.  4.  Reduce  71 9^  9s.  2d.  Canada  currency  to  Fe- 
deral money  ?  Ans.  gC877.83' 


10.  To  change  Federal  money  to  Canada  and  Nova- 
Scotia  currencies. 


♦  Note.— 5  shillings  of  this  currency  make  one  dollar.  The 
divisor  in  this  rule  is  obtained  by  reducing"  this  sum  to  the  de- 
cimal of  a  pound. 


184 


REDUCTION    OF    DECIMALS. 


Rule.  Multiply  the  given  sum  by  .25,  and  the  pro- 
duct will  be  pounds,  the  decimal  part  of  which  reduce  to 
shilliDgs  and  pence. 


EXAMPLES. 


Ex.  1,  In  68.5  dols.  how  much  Nova  Scotia  currency  } 
68.5 
.25 


3425 
1370 

17.125 
20 

£.500 
\t 

C.OOO 


Ans.  17/.  2s.  6c?. 


Ex.    2.    In  g  124.25  how  much  Canada  currency  ? 

Ans.  3U.    Is.  Sd, 

Ex.    3.    In   g7648    how    much  Nova    Scotia  cur- 
rency ?  Ans.  19I2Z. 

Ex.    4.    In  8867.35  ho        much  Nova  Scotia    cur- 
rency? Ans.  216/.  16s.  9d. 


Note. — The  shortest  method  of  working  the  exam- 
ples in  this  currency,  is  to  multiply  the  given  number  of 
pounds  by  4  for  dollars,  and  to  reduce  dollars  to  pounds 
divide  by  4. — This  number  is  used  because  5ihillingsare 
^^  or  1  of  a  pound. 


(  185  ) 


INVOLUTION, 


Involution  is  a  method  of  raising  numbers  to  higher 
powers. 

A  power  is  the  product  arising  from  multiplying  any- 
given  number  into  itself  once,  or  oftener  :  thus,  3x3 
ZZ  9  is  the  second  power  of  3,  and  it  is  denoted  in  thi» 
manner  3*. 

The  number  denoting  the  power  is  called  the  index^ 
or  exponent  of  that  power  :  thus,  in  3^  the  2  is  the  in- 
dex or  exponent. 

The  third  power  of  4-  is  4^  :^  4  X  4  X  4  ZI  64 
The  fourth  power  of  3  is  S*iz3x3X3X3:^8l 
The  sixth  power  of5is5«lz5x5x5x5X5x5 

[ZI  15625. 

The  third  power  of  i  is  |  ^  —  i  X  ^  X  J  ZI  ^V- 
The  fifth  power  of  .03  is  .03*  ZZ  .03  X  .03  X  .03  X  .03 
X  .03  =  .O0OOCUO243. 

EXAMPLES. 

Ex.  1.  What  is  the  sixth  power  of  6  ?     Ans.  46656 

2.  What  is  the  eighth  power  of  7  ?  Ans.  5764801 

3.  What  is  the  fourth  power  of  |  ?      Ans.  /^V 

4.  What  is  the  fifth  power  of  J.?      Ans.  i||gi 

5.  What  is  the  third  power  of  .25  ?  Ans.  ,015625 
0.  What  is  the  fourth  power  of  .05  ? 

Ans.  .00000625 


186  INVOLUTION. 

7.  What  is  the  third  power  of  .305  ? 

Ans.  .028372625 

8.  What  is  the  ninth  power  of  9  ? 

Ans.  S87420489 

9.  What  are  the  squares  of  3  and  6  ;    5  and  10 
6  and  12;  2,  4,  8,  and  16? 

Ans.     3*   =     9         52   =     25       6*   =     36 
6*    =  36       102   ^  100     122   ^  144 
22  =  4.       42  =  16     82  =      64     162  =  256 
10.  What  are  the  cubes  of  3  and  6 ;  5  and  10; 
6  and  12  ;  2,  4,  8,  and  16? 

Ans.  33-  =     .  27 

63  =       216  c=  8   X  27 

53  =   125 
103  =  1000  =  8  X  125  - 

63  =   216 
123  =  1728  =  8  X  216 

23  =  8 

43  =  64  =  8  X  8 

83  =  512  =  8  X  64 

163  -_  409a  =  8  X  512 


(187  ) 


EVOLUTION. 


Evolution  is  the  method  of  extracting  root9. 
The  root  of  any  number,  or  power,  is  such  a  number, 
as  being  multiplied  into  itself  once,  or  oftener,  produces 
that  power  :  thus  3  is  the  square  root  of  9,  because  3 
multiplied  into  itself  gives  9  :  4  is  the  cube  root  of  64, 
because  4  multiplied  into  itself  twice,  gives  64.  The 
roots  are  denoted  by  indices,  or  exponents,  in  this  man- 
ner: 

The  cube  root  of  125  is  -^^mIs  ~  5. 

The  square  root  of  81  is  ^  Slzz9, 

The  fifth  root  of  243   is  ^/Hs  zz  3. 

Ex.   1.  What  are  the  square  roots  of  49  and  64  ? 

Answer,  7.  8. 

2.  What  are  the  cube  roots  of  216,  343,  512,  and 
729  ?  "  Answer,  6.  7.  8.  9. 

3.  What  are  the  fourth  roots  of  625,  2401 ,  and 
4096  ?  Answer,  5.  7.  8. 

4.  What  are  the  fifth  roots  of  3125  and  32768  ? 

Answer,  5.  8 

To  extract  the  square  root. 

Rule.  (1.)  Divide  the  given  jumber  into  periods  of 
two  figures  each,  by  placing  a  dot  over  units,  another 
over  hundreds,  and  so  on.  {2.)  Find  the  greatest  square 
in  the  first  period,  and  set  its  root  on  the  right-hand,  as  a 
quotient  figure  in  division.  (3  )  Subtract  the  square 
thus  found,  and  to  the  remainder  annex  the  succeeding 
period  for  a  new  dividend.  (4.)  Double  the  root  for  a 
divisor,  and  examine  how  often  it  is  contained  in  the 
dividend,  exclusive  of  the  place  of  units,  and.put  the 


■«*. 


188  EVOLUTION. 


result  into  the  quotient  and  in  the  units  place  of  the  divi- 
sor. (5  )  Multiply  the  divisor  thus  increased  hy  the 
new  quotient  fig^ure,  and  subtract  the  product  from  the 
dividend.  (t>.)  Bringdown  the  next  period,  find  a  divi- 
sor as  before,  by  doubling  the  figures  already  in  the  root, 
and  proceed  as  before. 

The  rule  will  be  rendered  clear  by  the  follcvin^  examples: 

What  are  the  square  roots  of  16777216  and  43040321  ? 

167771^10(4096 
16 

809).  .7772 
7281 

8186)49:16 
49116 


43046721(6561 
36 


125)704 
6-25 


1306):"967 
7836 


13121)13121 
13121 

EXAMPLES. 

Ex.  1 .  What  is  the  square  root  of  1 1 7649  } 

'     Answer,  343. 

2.  What  is  the  square  root  of  262144  ? 

Answer,  512. 

3.  What  is  the  square  root  of  531441  ? 

Answer,  729. 

4.  What  is  the  square  root  of  1679616  ? 

Answer,  1296, 


EVOLUTION.  18^ 

Ex.  5.  What  is  the  square  root  of  5764801  ? 

Answer,  2401. 

6.  What  is  the  square  root  of  J  073741 824.  ? 

A;)9wer,  32768. 

7.  What  is  the  square  root  of  1 195506G9121  ? 

Answer,  345761. 

8.  What  is  the  square  root  of  20  ? 

Answer,  4.4721,  &c. 

9.  What  is  the  square  root  of  300  ? 

Answer,  17.3205,  &c. 

10.  What  is  tlie  square  ro©t  of  1000  ? 

Answer,  S1.622,  &c. 

1 1 .  What  is  the  square  root  of  | :  -^^ ;  -^^  ?     ♦ 

Answer,  .7071,  &c.  .4082,  &c.  .4166. 

12.  What  is  the  square  root  of  .25  ? 

Answer,  .5. 

MISCELLANEOUS    EXAMPLES. 

Ex.  1.  A  gentleman  desirous  of  making  his  kitchen 
garden,  which  is  to  contain  4  acres,  a  complete  square, 
I  demand  what  will  be  the  length  of  the  side  of  the 
garden  ?  Ans.  139  yards. 

Ex.  2.  Six  acres  of  ground  are  to  be  allotted  to  a 
square  garden  ;  but  for  the  sake  of  more  wall  for  fruit, 
there  is  to  be  a  smaller  square  within  the  lar;>;er,  which 
is  to  contain  3  acreSj  1  demand  the  length  of  the  sides 
of  each  square  ?  Ans.  ouJ;er  170.41  yds.  inner  120^nearlj. 

Ex.  3.  What  is  the  mean  proportional  between  12 
and  75  ?  Ans.  30. 

Kx.  4.  flow  long  must  a  ladder  be  to  reach  a  winilow 
30  feet  high,  when  the  bottom  stands.  12  feet  from  the 
house  ?  Ans.  22.31  feet. 

To  extract  the  cube  root, 

I.  UuLE.  (1)  Find,  by  trials,  the  nearest  cube  to 
the  given  number,  and  call  it  the  assumed  cube.  (2) 
Say  as  twice  the  assuuied  cube  added  to  the  given  num- 
ber, is  to  twice  tiie  number  added  to  the  assumed  cube, 
80  is  the  root  of  the  assumed  cube  to  the  root  required 
nearly. 


190  EVOLUTION. 

What  is  the  cube  root  of  27455  ? 

Here  the  nearest  root  that  is  a  whole  nnmber  is  3<^, 
the  cube  of  which  is  27000  :  therefore  I  say, 

As  27000  X  "2  +  2745.5  :  27456  X  2  +  27OO0  :  :  30 
or  81455  :  8l9lO  ::    30  :  30.1675. 

It  is  evident  that  the  true  root,  omitting  the  last  two 
figures,  is  somewhere  between  30.16  and  30.17,  the  for- 
mer being  too  little,  the  latter  something  too  large.  By 
taking  the  root  thus  found  30.16,  as  the  assumed  eube, 
and  repeating  the  operation,  the  root  will  be  had  to  a  still 
greater  degree  of  exactness, 

Ex.  1.  What  is  the  cube  root  of  15625  ?       Ans.  25 

2.  What  is  the  cube  root  of  140608  ?    Ans.  52 

3.  What  is  the  cube  root  of  444194947  ?  Ans.  763 

4.  What  is  the  cube  root  of  the  difference  be- 
tween 140608  and  14625  ?  Ans.  50.13  nearly. 

II.  Rule.  (1)  Separate  the  given  number  into  periods 
of  three  figures  each,  beginning  from  units  place  ;  then 
from  the  first  period  subtract  the  greatest  cube  it  con- 
tains, put  the  root  as  a  quotient,  and  to  the  remainder 
bring  down  the  next  period  for  a  dividend.  (2)  Find 
a  divisor  by  multiplying  the  square  of  the  root  b}"^  300, 
see  how  often  it  is  contained  in  the  dividend,  and  the 
answer  gives  the  next  figure  in  the  root.  (3)  Multiply 
the  divisor  by  the  last  figure  in  the  root.  Multiply  all 
the  figures  in  the  root  by  30,  except  the  last,  and  that 
product  by  the  square  of  the  last.  Cube  the  last  figure 
in  the  root.  Add  these  three  last  found  numbers  to- 
gether, and  subtract  this  sum  from  the  <lividend  ;  to  the 
remainder  bring  down  the  next  period  for  a  new  divi- 
dend, and  proceed  as  before. 


EVOLUTIONS.  191 

Ex.  5.  What  is  the  cube  root  of  444194947  ? 


444194947(763  Answer. 
343 


7  X  7  X  300  =  14700)'Oi  194 
95J76 


76x76X3O0  =  l7328<)O)5'218^;4r  1732800  =  divisor 

14.700  ==  divisor       5'^l<3i7  3 

5198400 

8821.0  20520=76X30X9 

7560=7X30X36  27=  3  X  3  X3 

216=6X6X6  


95976 


52J8947 


Ex.  6.  What  is  the  cube  root  of  46656  ?       Ans.  36 

7.  What  is  the  cube  root  of  65939.264  ?  Ans.  404 

8.  What  is  the  cube  root  of  3,  carried  to  2  places 
tf  decimals  ?  Ans-  1 .44 

9.  What  is  the  cube  root  ofy|^  ?  Ans.  J 

10.  What  is  the  cube  root  of  r^^j  ?       Ans.  ^ 

11.  What  is  the  cube  root  of  .729  ?         Ans.  .9 

12.  What  is  the  cube  root  of  .003375  ?  Ans.  .15 

MISCELLANEOUS  EXAMPLES. 

Ex.  1.  What  is  the  length  of  one   side  of  a  vessel, 
which  contains  13824  solid  inches  ?    Ans    24  inches. 

Ex.  2.  In  a  cubical  building  that  measures  2744  feet, 
what  is  the  length  of  a  side  ?  Answer,  1 4 


19S  ARITHMETICAL  PROGRESSION. 


ARITHMETICAL  PROGRESSION. 


When  a  series  of  numbers  increases  or  decreases  by 
some  common  excess,  or  common  difference,  it  is  said  to 
be  in  arithmetic^  progression,  such  as  I,  3,  5,  7,  9,  &c., 
and  12,  10,  8,  6,  i,  ^c. 

The  nuMibei  s  which  form  the  series  are  called  the  terms 
©f  the  progression  ;  of  these  the  fir&t  and  last  are  called 
the  extremes. 

The  first  term  is   called         -         -  a 

The  last  term  is  called  -         -  « 

The  number  of  terms  is  called        -  n 

The  conimon  di'Terence  is  called     -  d 

The  sum  of  all  tue  terms  is  called  s 

Any  t))ree  of  these  terms  being  given,  the  others  may 
be  easily  found. 

T.  When  the  first  term  a,  and  the  last  term  z,  and 
the  number  of  terms  w,  are  given,  to  find  the  sum  of  all 
the  terms,  s. 

Rule.  Multiply  the  sum  of  the  extremes  by  the  num- 
ber of  terms,  and  divide  by  2,  the  quotient  is  the 
answer  :  or 


fl-f  z  X — 3:s. 
2 
Ex.  I.   What  is  the  sum  of  an  arithmetical  series, 
whose  first  terra  is  5,  last  term  29,  and   the  number 
terms  7. 

7  7        238 

Here  s=5-|-29x — ==34x— = =  119,  the  answer. 

2  3  2 


1 


^1 


ARITHMETICAL  PROGRESSION.  19:^ 

Ex.  2.  The  first  and  last  terms  of  a  series  are  3  and 
111,  and  the  number  of  terms  37  :  what  is  the  sum  ? 

Ans.2109 

Ex.  3.  How  many  strokes  do  the  clocks  of  Venice 
strike  in  24-  hours,  where  they  strike  from  1  to  24-  ? 

Ans.  300 

Ex.  4.  The  first  and  last  terms  of  a  series  are  1  and 
1000,  and  the  number  of  terms  100  :  required  the  sum. 

Ans.  50050 

Ex.  5.  If  100  stones  are  placed  in  a  ripjht  line,  exactly 

a  yard  asunder,  and  the  first,  one  yard  from  a  basket, 

what  length  of  ground  will  a  man  go  over,  who  gathers 

them  up,  one  by  one,  returning  with  each  to  the  basket  ? 

Ans.  5  miles  and  1300  yards. 

Ex.  6.  What  must  a  man  give  for  54  timber  trees,  for 
which  he  pays  5  shillings  for  the  first,  and  QOl.  for  the 
last,  and  the  prices  of  the  others  being  in  arithmetical 
progression  ?  Ans.  546/.  15s. 

Ex.  7.  A  butcher  buys  a  drove  of  oxen,  consisting  of 
32  ;  for  the  first  he  pays  15s  ;  and  for  the  others  he  is  to 
pay  in  arithmetical  progression,  so  that  for  the  last  he  is 
pay  38i. :  what  will  they  all  come  to.  Ans.  620^ 

Ex.  8.  A  horse-dealer  sends  to  a  fair  63  horses  of  va- 
rious kinds  and  worth,  which  he  is  willing  to  dispose 
of  according  to  the  principles  of  arithmetical  progression, 
demanding  3/.  only  for  the  first,  provided  he  nad  53^  for 
the  last:  how  much  did  he  receive  for  the  whole,  and 
what  was  the  average  value  of  each  horse  ? 

Ans.  J  7  64/.  for  the  whole*— 28^,  average  price  of  each 
horse. 

II.  The  first  and  last  terms,  a  and  Zy  and  number  of  terms 
being  given,  to  find  the  common  difference  d. 

Rule.  The  difference  of  the  extreme  terms  divided 
by  the  number  of  terms  less  1,  will  be  the  common  dif- 
ference sought : 


or =d, 

n — 1 


V 


17  .-^    -  7 


I94f  ARITHMETICAL    PROGRESSION. 

Ex.  1 .  What  is  the  common  difference  of  an  arithme- 
tical progression,  whose  extremes  are  8  and  200,  and  the 
number  of  terms  17  ? 

8^200        200—8        192 

16  16  16 

Ex.  2,  When  the  extremes  of  an  arithmetical  progres- 
sion are  6  and  57,  and  the  number  of  terms  18,  what  is 
the  common  difference  ?  Ans.  3 

Ex,  3.  A  gentleman  gives  at  Christmas,  among  his  25 
poor  neighbours,  a  sum  of  money  in  arithmetical  progres- 
sion •:  to  the  least  needy  he  gives  5  shillings,  and  to  the 
poorest,  with  a  very  large  family,  he  gives  five  guineas  : 
what  was  the  common  difference  ?  Ans.  4s.  2d, 

Ex.  4.  A  traveller  is  out  on  his  journey  a  month,  of 
which  he  travels  25  days  ;  on  the  first  he  rides  7  miles, 
and  on  the  last,  having  little  to  do,  he  comes  43  miles  : 
how  much  was  the  daily  increase  of  his  travelling,  and 
how  many  miles  did  he  ride  in  the  whole  ? 

Answer  1 J  miles  increase — 625  the  number  of  miles 
travelled. 

III.  The  extreme  terms  a  and  z,  and  common  difference 
d  being  given,  to  find  the  number  of  terms  r. 
Rule.  Divide  the  diff*erence  of  the  extremes  by  the 
common  difference,  and  the  quotient  increased  by  unitj 

is  the  number  sought  :  ^^—^ |-Iizn. 

Ex.  1.  When  the  extremes  are  4  and  106,  and  th^ 
common  difference  is  3,  what  is  the  number  of  terms  ? 
4w?^l06  106—- 4  102 

\-\zi' 1-1=: +ir=34+i:=35z^ft. 

3  3  S 

Ex.  2.  If  the  least  term  be  6,  the  greatest  216,  and 
the  common  difference  5,  what  is  the  number  of  terms  ? 

Ans.  48 
Ex.  S.  What  debt  can  be  paid,  and  in  what  time^  sup- 
posing I  agree  to  lay  by  3s.  the  first  week,  7s.  the  next, 
lis.  the  thir*,  and  soon  in  arithmetical  progression, 
till  the  last  saving  be  four  guineas  ?  Ans.  46^  4s  4^d.^t 
the  debt  to  be  paid.^21^  weeks=the  time. 


^K 


GEOMETRICAL    PROGRESSION; 

\ 

Ex.  4.  I  set  out  for  Hastings,  which  is  69  miles  from 
this  place,  and  I  walk  the  first  dayj4  miles,  the  second  7, 
ilicreasing  every  day  by  3  miles,  and  on  the  last  19  miles : 
how  many  days  will  the  journey  take  ? 

Ans.  6,  the  number  of  day's  journey. 

In  addition  to  the  above,  the  learner  may  commit  to 
memory  the  following  facts  on  the  subject : 

1.  If  three  numbers  are  in  arithmetical  progression, 
the  sum  of  the  extremes  is  equal  to  double  the  mean 
term;  as,  6,  9,  12,  where  6 -f- 12=2 x9~  18. 

2.  If  four  numbers  be  in  arithmetical  progression,  the 
sum  of  tlie  two  extremes  is  equal  to  the  sum  of  the 
means;  as  5,  8,  11,  14,  where  5  +  14.zi8-hIlZll9. 

S.  When  the  number  of  terms  is  odd,  the  ^double  of 
the  middle  term  will  be  equal  to  the  sum  of  the  ex- 
tremes :  or  of  any  other  two  means  equally  distant 
from  the  middle  term;  as  3,  8,  13,  18,  23,  28,  33, 
where  3+33— 2x18  =  13+23=8+28. 


GEOMETRICAL  PROGRESSION. 


A  Geometrical  Progression  is  a  series  of  numbers, 
the  terms  of  which  gradually  increase  or  decrease  by 
the  constant  multiplication  or  division  of  some  particu- 
lar number;  as  1,  3,  9,  27,  81,  243,  &c.,  or  64,  32, 
16,  8,  4,  2,  1,  i,  &c. 

In  the  first  case,  the  series  is  increasing  by  the  con- 
stant multiplication  of  3;  in  the  second,  it  is  a  decreas- 
ing series  by  the^  constant  division  of  2v  It  is  evideni: 
that  both  series  1|»ay  be  carried  on  for  ever. 

The  number  by  which  the  series  is  constantly  increas- 
ed or  diminished  is  called  the  ratio. 

The  first  term  is  called      *    -    -  -  a 

The  last  term  is  called      ~    .    •  .  z 

The  number  of  terms  is  called    -  •  w 

The  ratio  is  called         ...    -  -  t* 

The  sum  of  all  the  terms  is  called  -  s. 


196  GEOMETRICAL  PROGRESSION. 

Any  three  of  thesft  terms  being  given  or  known,  tiie 
others  may  be  determined. 

1.  Given   the  first  term   a,    the   last  term  z,  and  the 

common  ratio  r,  to  find  the  sum  s. 
Rule.     Multiply  the  last  term  by  the  ratio,  and  from 
the  product  subtract  the  first  term,  and  the  remainder 
divided  by  the  ratio,  less  one,  will  give  the  sum  of  the 
s«ries;  or  xxr — a 

=s. 

r—\ 
Ex.  1.  The  first  term  of  a  series  in  geonietrical  pro- 
gression is  5,  the  last  term  is  3645  and  the  ratio  3f 
what  is  the  sum  J  ' 

3645x3—5     10935—5     10950 

Here  s= = zi =5^65» 

3—1  2  2 

For  the  terms  are  5,  15,  45,  135,  405,  1215,  and 
^645  5  which,  being  added  together,  make  5465. 

Bx.  2.  The  first  and  last  terms  of  a  geometrical  se- 
ries are  4  and  3294172,  and  the  common  ratio  is  7  : 
what  is  the  sum  ?  A.  3843200 

Ex.  3.  The  first  and  last  terms  of  a  geometrical  pro- 
gression are  4  and  262144,  and  the  ratio  4:  what  is  the 
sum.^  A.  349524 

T.  Given  the  first,terma,  the  number  of  terms   «,  and 
the  ratio  r,  to  find  the  last  term  z. 

The  last  term  n^y  be  obtained  by  continual  multiplica- 
tion ;  but  as  that,  in  a  long  series,  is  a  tedious  process, 
we  shall  give  the  following  rule  : 

1.  When  the  first  or  least  term  is  equal  to  ratio. 

Rule.  Write  down  some  of  the  leading  terms  of  the 
geometrical  §eries,  over  which  place  the  arithmetical  se- 
ries 1,  2,  3,  4,  &c.,  as  indices;*  find  what  figures  of 

•  When  the  natural  numbers  1,  2,  3,  4,  5,  &c.,  are  set  over  a 
geometrical  series,  they  are  called  itidices  or  exponents,  and 
they  shew  the  distance  of  any  term  from  unity,  or  from  the  first 
term  :  thus,  in  the  series  2',  4%  83,  16*,  64^,  128S  &c.,  1.  2, 
3,  &c.  are  the  indices,  and  shew  the  distance  of  any  term  ;  .the 
series  from  the  first  term,  index  5,  for  instance,  shews  that  64  is 
the  fifth  term  in  the  series. 


GEOMETRICAL    PROeRESSION.  I9f 

these  indices  added  together  will  give  the  index  of  the 
term  wanted  in  the  geometrical  series  ;  then  multiply 
the  numbers,  standing  under  such  indices,  into  each 
other,  and  their  product  will  be  the  term  sought. 

Ex.  1 .  What  is  the  last  term  of  a  geometrical  series 
"having  13  terms,  of  which  the  first  is  2,  and  the  ratio  2jf 

Here  the  series,  with  their  indices,  will  stand  thus ; 
2S  42,  83,  164,  325,  64.6,  &c. 

The  number  of  terms  being  13,  the  index  to  the  last 
term  will  be  13  equal  to  the  indices  2+5+6,  which  fi- 
gures standing  over  4,  32,  and  64,  shew  that  these  last 
are  to  be  multiplied  together,  and  the  product  is  the 
term  sought;  thus  4x32x64=8192. 

Ex.  2.  What  is  the  last  term  of  the  series  having  9 
terms,  of  which  the  first  is  3,  and  the  ratio  3  ? 

Answer,  19683. 

Ex.  3.  What  did  the  last  of  IS  oxen  cost,  the  first  of 
which  was  sold  for  3s. ;  the  second  for  9s.  and  so  on. 

Answer,  265721.  Is, 
2.  When  the  first  term  a,  of  the  series,  is  not  equal  to 
the  ratio  r. 

Rule.  Write  down  the  leading  terms  of  the  serie^ 
and  place  their  indices  over  them,  beginning  with  a  cy- 
pher, add  together  the  most  convenient  indices  to  make 
an  index  less  one  than  the  number  expressing  the  place 
of  the  term  sought;  then  multiply  the  numbers  stand- 
ing under  such  indices,  into  each  other,  dividing  the 
product  of  every  two  by  the  first  term  in  the  geometri- 
cal series  ;  the  last  quotient  is  the  term  required. 

Ex.  1.  What  is  the  last  term  of  the  series,  whose 
first  term  is  4,  ratio  3,  and  numbers  of  terms  15  ? 
40,  12S  362,  ,083,  3244^  97^5,  29i6S  &c. 

The  number  of  terms  being  15,  the  index  sought  must 
be   14  equal  to  6+5+3,  under  which  stand  the  terms 
2916,  972,  and  108,  then 
2916  X  972  708588  X  108 

=708588,  and =  19131876  = 

4  4 

r  =  last  ternv 

17* 


198  GEOMETRICAL     PROGRESSION. 

Ex.  2.  The  first  term  of  a  geometrical  series  is  2,  the 
•number  of  terms  12,  and  the  ratio  5,  required  the  last 
tferm  ?  Answer,  97656250 

Ex.  3.  The  first  term  of  a  geometrical  series  is  1,  the 
ratio  2,  and  the  number  of  terms  25,  what  is  the  last 
term,  and  also  the  sum  of  all  the  terms  ? 
•  Answer,  16777216  last  term     S3554431  sum. 

^  Ex,  4.  The  first  term  of  a  series  is  5,  the  ratio  3,  and 
the  number  of  terms  16,  what  is  the  last  term,  and  the 
sum  of  the  terms  ? 

Answer,  7 1744535  last  term     107dl  6800  sum. 

Ex.  5-  A  hosier  sold  12  pair  of  stockings,  the  first 
pair  at  Sd.,  the  second  9c?.,  and  so  on  in  •geometrical 
progression :  for  what  did  he  sell  the  last  pair,  and  how 
much  had  he  for  the  whole  ? 

Answer,  531441  last  term,  5S2\l    10s.  sum. 

Ex.  6.  What  would  a  horse  fetch,  supposing  it  was 
sold  on  condition  of  receiving  for  it  one  farthing  for  the 
first  nail  in  his  shoes,  a  halfpenny  for  the  second,  one 
penny  for  the  third,  and  so  on,  doubling  the  price  of 
every  nail  to  32,  the  number  in  his  four  shoes  ? 

Answer,  4473924/.  5s.  S|rf. 

Ex.  7.  A  husbandman  agreed  to  serve  his  master  dur- 
ing hay -time  and  harvest,  or  five-and-foity  clear  days, 
provided  he  would  give  him  a  barley-corn  only  for  the 
first  day's  work,  3  for  the  second,  9  for  the  third,  and  so 
on  in  geometrical  proportion ;  what  would  he  have  to 
receive  in  money  for  his  labours,  supposing  there  were, 
half  a  million  of  grains  in  a  bushel,  and  each  bushel 
was  worth  4s.  ? 

Ans.   590-S62.54l.3l0.166/.   14s.  9|g?.  /o^o* 

"IChe  following  facts  may  be  committed  to  memory : 

1.  If  three  numbers  are  in  geometrical  progression, 
the  product  of  the  extremes  is  equal  to  the  square  of  the 
mean ;  as^,  9,  27,  here  3  x  27  =  9  x  9  =  81. 

2.  If  four  numbers  are  in  geometrical  progression,  the 
product  of  the  extremes  is  equal  to  the  product  of  the 

8,  16  5  here  2  X  16  =«  4  X  8  =  32i 


INTEREST.  199 

3.  If  the  series  contain  an  odd  number  of  terms,  the 
squ'dre  of  tl^e  middle  t"rm  is  equal  to  the  product  of  the 
adjoining  extremes,  or  of  auv  two  terms  eo'ia'ly  <1istant 
from  the  n  ;  as  3,  9,  27,  81,'  243  ;  here  27 ^  =  3  X  243 
=  9  X  81  =  729. 


INTEREST. 


Interest,  is  the  sum  of  money  paid,  or  allowed  for 
the  loan  or  use  of  some  other  sum,  lent  for  a  certain 
time,  according  to  a  fixed  rate. 

The  sum  lent,  and  on  which  the  interest  is  reckoned 
is  called  the  Prinoipal. 

The  sum  per  Cent,  agreed  on  as  interest,  is  called  the- 
Rate. 

The  principal  and  interest  added  together,  is  called 
the  Amount. 

Interest  is  distinguished  into  Simple  and  Compound. 

Simple  Interest,  is  that  which  is  reckoned  on  the 
principal  only,  at  a  certain  rate  for  a  year,  and  at  a  pro- 
portionately greater  or  less  sum,  for  a  greater  or  less 
term  :  thus,  if  5^  is  the  rate  of  interest  of  100/.  for  one 
year,  loL  is  the  interest  for  two  years,  20/.  for  four 
years,  and  so  on. 

Rule.  (1)  Multiply  the  principal  by  the  rate,  and 
divide  the  product  by  100,  and  the  quotient  is  the  inte- 
rest for  one  year. 

250X5 

Thus  the  interest  of  250/.,  at  5  per  cent,  is*-.- :^ 

100 
12/.   lOs. 

(2)  Multiply  the  interest  for  one  year  by  the  number 
of  years,  and  the  product  is  the  interest  for  the  same  : 


200 


INTEREST. 


Thus  the  interest  of  250^  for?  years  is  12Z-  lOs.  x  T 

=t  m.  los. 

(3)  If  parts  of  a  year  be  given,  they  must  be  worked 
for  by  the  aliquot  parts  of  a  year,  as  m  Practice,  or  by 
the  Rule  of  Three  Direct. 

Ex.  1.  What  is  the  interest  of  853^  10s.  for  4  years 
and  8  months,  at  5  per  cent,  per  annum  ? 

42  13  6  =  interest  for  one  year. 
4 


853  10 
5 

Ir.  42.67   10 
20 


170   14  0 

21      6  9 

7     2  3 


shill.  13.50 
12 


L.  199     3  0 


pence6.00 


199/.     3s.     Od, 


Answer, 

To  find  the  amount,  I  must  add  the  principal  to  the 
interest.  In  this  example,  the  amount  is  equal  to  853Z. 
10s.  -h   l99i.  3s.  =   10521.  13s. 

Ex.  2.  What  is  the  amount  of  142^    lOs.  for  four 
years  and  52  days  at  4J  per  cent  ? 
L.  142  10        L.     s.    d, 

4^       6     8      3=  interest  for  one  year. 


570 
71 

0 
5 

i. 

6.41 
20 

5 

shill. 

8.25 
12 

pence 

3.00 

'25  13    Ost=interest  for  four  years. 


To  find  the  interest  for  the  52  daj^ 
I  sav. 


INTEREST. 

days.     L.  s.  d.    days. 

If  305  :   6     8    3::  5^^. 

20 

L. 

s. 

d. 

25 

•3 

0                          12B 

0 

18 

S^                           12 

26 

11 

3i=interest      1539 

142 

10 

P  ^principal       52 

169 

1 

3i-=amount      3078 

7695 

12 

365)80028(2191" 

201 


1  s  3J  =interest  for  52  ds, 
Ex.  3.  What  is  the  interest  of  46 U.  at  4  per  cent, 
for  5  years?  Ans.  92/.  3s.   \\\d. 

Ex.  4.    What  is  the  interest  of  230Z.    15s.  for   6-^ 
years,  at  5  per  cent,  per  annum  ?  Ans.  74/.  19s.  lO^i. 
Ex.  5.   What  is  the  amount  of  225/.  for  7  years,  at 
3i  per  cent,  per  annum  ?  Ans.   280/.  2s.  6^. 

Ex.  6.  How  much  shall  I  have  to  receive  at  the  end  of 
5  years  for  350/.  supposing  4^  per  cent,  be  allowed  as 
interest  ?  Ans.   428/.   15s. 

In  most  computations  relating  to  simple  interest,  the 
work  is  shortened,  if  the  interest  of  l/.  for  a  given  term 
is  known,  as  the  interest  of  any  other  sum  for  the  same 
term  will  then  be  found  by  only  multiplying  by  the 
given  sum. 

The  interest  of  1/.  for  a  year  must  be  in  the  same 
proportion  as  the  interest  of  lOO/.  to  its  principal ; 
therefore,  at  5  per  cent ,  we  say,  as  100/.  :  5/o  :  :  1/.  ' 
.05/.     Hence  the  interest  of  1/.  for  one  year, 

/..  L, 

At  3  per  cent,  is        -         -        -        ,03     • 

3i ,035 

4 ,04 

4A       -         -  -         -_        -  .045 

5         -         -  -         .         -         ,05 


202  INTEREST. 

Ex.  7.    What  is  the  interest  of  540  dollars  for  1 
^ear,  at  6  percent,  per  annum  ?  Ans.  §32.40 

Ex.  8.  What  is  the  interest  of  §275.50  for  3  years 
at5iper  cent,  per  annum  ?         Ans.  §45. 45c.  7.5m. 

Ex.  9.  What  is  the  interest  of  §1034.25  for  4  years 
at  6^  per  cent,  per  annum. ^         Ans.  S258.56c.  2.5m. 

Ex.  10.  How  much  will  750  dollars,  amount  to  in  7|. 
years  at  5|  per  cent,  per  annum  ? 

Answer,  81073.43cts.7.5  millg.* 


INTEREST. 


Tbe  hiterest  of  One  Pound  for  any  number  of  Years. 


Years. 

3  per 
Cent. 

10 

MJ 

20 

,6 

30 

,9 

40 

1.2 

50 

1,5 

60 

1,8 

70 

2,1 

80 

2,4 

90 

2,7 

100 

3,0 

S^  per 
Cent. 


,33 

,7 

1,05 

1,4 

1,75 

2,1 

2,45 

2,8 

3,15 

3,5 


4  per 

4i  per 

iper 

Cent. 

Cent 

Cent. 

.4 

.45 

,5 

,8* 

,9 

1,0 

1,2 

1,35 

1,5 

1,6 

1,8 

2,0 

2,0 

2,25 

2.5 

2.4 

2J 

3,0 

2,8 

3,15 

3,5 

3,2 

3,6 

4,0 

3,6 

4,05 

4,5 

4,0 

4,5 

5,0 

The  365th  part  of  the  yearly  interest  is  always  considered  as  the 
proper  interest  for  a  day,  and  its  multiples  as  the  interest  for  any 
number  of  days  ;  thus,  at  5  per  cent,  the  interest  for  a  daj  is 

-.05 

=.00013o9  ;   and  the  interest  for  12  days,  at  the  same  rate, 

365 

is  .0t0;369  X  li=:.0016428.  Hence  by  means  of  the  following 
table,  all  calculations  at  5  per  cent.  Simple  Interest  are  easily 
performed,  for  any  number  of  days. 


days 

Interest 

days 
26 

interest.  |days 

Interest. 

days 

Interest. 

1 

,0U0156y 

,0U3n616 

51 

,0069863 

76 

,0104109 

2 

.0002739 

27 

0036986 

52 

,0071232 

77 

,0105479 

3 

,0004109 

2« 

,0038336 

53 

,0072602 

78 

,0106849 

4 

,0005479 

29 

,0039726 

54 

,0073972 

79 

,0108219 

5 

,0006^49 

30 

,0041095 

55 

,0C7..342 

80 

,0109589 

6 

,0008219 

31 

,0042465 

56 

,0076712 

81 

,0110958 

7 

,0009589 

32 

,0043835 

57 

,007«082 

82 

,0112328 

8 

,0010958 

33 

,0045205 

58 

,0079452 

83 

,0113698 

9 

,0012328 

34 

,0045575 

59 

,0080821 

84 

,0^15068 

10 

,0013698 

35 

,0047945 

60 

,0082191 

85 

,0116438 

11 

,0015068 

36 

,00^93 1 5 

6] 

,0083561 

86 

,0117808 

12 

,0016438 

37 

,0050684 

62 

,008493 1 

87 

,0119178 

13 

,0017808 

t8 

,0052054 

63 

,0086301 

88 

,0120547 

14 

,6019178 

39 

,0053424 

64 

,00876? 1 

89 

,0121917 

15 

,0020547 

40 

,00547^^* 

65 

,0089041 

90 

,0123287 

16 

,0021917 

41 

,0056164 

66 

,0090411 

91 

,0124657 

17 

,0028287 

42 

,0057534 

67 

,0091780 

92 

,0126027 

18 

,0024657 

43 

,0058904 

68 

,0093150 

93 

MV27397 

1^ 

,0026027 

44 

,0060274 

69 

,00a4:>20 

94 

.  12i767 

30 

,0027397 

45 

,0061643 

70 

,00958i^0 

95 

,0130137 

21 

,0028767 

46 

,0063013 

71 

,0097i>60 

96 

,0131506 

22 

,00.30137 

47 

,0064383 

72 

,009863> 

97 

,0132876 

23 

,0031506 

48 

,0065753 

73 

,01000CO 

98 

,0134246 

24 

,0032876 

49 

,0067121 

74 

,0101369 

99 

,0  35616 

25 

,0014246 

50 

,0068493 

75 

,0102739 

100 

,0136986 

204  COMMISSION    AND    BROKERAGE. 

Rule.  Multiply  the  figures  corresponding  with  the 
number  of  days  by  the  sum  : 

Thus,  if  the  interest  of  75l.  for  61  days  be  required  :  I 
find  opposite  to  61,  the  number  .0083561,  wliich  multi- 
plied by  75,  gives  .6267075  of  a  pound,  which  reduced, 
is  iQs.aid. 

Ex.  I .  What  is  the  interest  of  155Z.  for  49  days  ? 

Ans.  1^  Os.  9|rf.  nearly 


COMMISSION  AND  BROKERAGE. 


Commission  is  an  allowance  of  a  certain  sum  per  cent, 
to  a  correspondent  or  agent,  for  buying  and  selling  goods 
for  his  employer,  or  to  a  banker  {or  drawing  bills  and 
managing  accounts. 

Broker  A«E,  though  of  a  difierent  name,  is  of  the  same 
nature  as  Commission. 

Ex.  1.  A  salesman  a^  Smithfield,  in  the  course  of  a 
year,  sells  for  his  correspondents  1 120  loads  of  hf.y,  at 
the  average  price  of  5/.  lOs.  per  load  ;  and  620  loads  of 
straw,  at  55s.  per  load  :  1  wish  to  know  t!.e  c  .mmission 
money,  at  ^i  per  cent  .^  Answer,  176/.  1 9s.  3d, 


COMMISSION    AND  BROKERAGE.  205 

1120  620 


H 

= what  the  hay 

sold  for« 

m 

5600 
560 

L.6l60= 
1705 

L.7865 
2i 

1240 
310 
155 

1705  = 

:  what  the  straw 
[sold  foi;, 

15730 
1966.25 

176.9625 
20 

19.25 
12 


Answer,  176^.  i9s.  Sd. 


3.00 

Ex.  2.  A  Manchester  manufacturer  allows  his  agent 
in  London  i-}  per  cent,  for  goods  sold  by  him  ;  in  the 
course  of  the  year  1807  he  sold  to  the  amount  of  15,400/., 
what  was  his  commission  for  that  year,  an^  how  much 
was  the  agent's  clear  gains,  supposing  his  losses  on  the 
year's  account,  by  bad  debts,  amounted  to  225/.  10s.  6d,  ? 
Ans.  654/.  10s.  Od.  Com.  428/..l9s.  brf.  clear  gains 
Ex.  3.  A  Liverpool  merchant  sells  goods  in  a  year, 
for  his  American  correspondents  to  the  amount  of 
144,454/.  lOs.,  on  which  he  reckons  his  clear  gains  at  the 
rate  ot  |  per  cent.,  what  is  his  income  on  this  one  concern? 

Answer,  54  iZ.  14s.  Id 

Ex.  4.  What  is  the  commission  of  g  1026.50,  at  3  J  per 

cent  ?  Ans.  38  dolls.  49  cts.  3.75  mis. 

Ex.  5.  A  bookseller  in  London  allows  his   agent  in 

America  5  per  cent,  commission  ;  what  does  he  pay  him 

for  the  remittance  of  8540i,  15s  9d,  ? 

Answer,  427/.  Os.  9ld, 
18 


20d 


BISCOUNT. 


Ex.  6.  What  is  the  brokerage  of  gl210,  at  ^  per 
cent.  ?  Answer,  3  dolls.  2  cts.  5  mis. 

Ex.  7.  What  is  the  claim  of  a  broker  at  3|  per  cent. 
on  gl550.50.  ?  Ans.  52  dolls.  32  cts.  9.375  mis. 

Ex.  8.  What  is  the  commission  on   glOOO  atf  per 
cent.  ?  Answer,  6  dolls.  25  cts. 

Ex.  9.  What  have  I  to  pay  my  broker  for  the  sale  of 
goods  to  the  amount  of  9950/.  95.,  at  li  per  cent.? 

Answer,  124/.  7s.  7^d. 

Ex.  10.  What  will  the  commission  of  a  country  banker 
amount  to  on  12314i.  8s.  9d.,  at  |  per  cent.  ? 

Answer,  15/.  7s.  lO^dr. 

Ex.  II.  What  is  the  brokerage  of  1526/.  iSs,  6d,,  at 
1^  per  cent.  ?  Answer,  22/.  I8s. 


DISCOUNT. 


Discount  is  an  allowance  made  for  advancing  mo- 
ney on  securities  before  they  are  due.  The  presen^t 
worth  of  any  sum,  due  sometime  hence,  is  such,  as  if 
put  to  interest  for  that  time  at  the  rate  per  cent,  given, 
would  amount  to  the  given  sum. 

Rule.— As  the  amount  of  100/.  or  dollars,  at  the 
rate  and  time  given  is  to  KfO  :  so  is  the  given  sum  to  the 
present  worth.  The  present  worth  taken  from  the 
given  sum  will  be  the  rebate  or  discount. 

or  thus,  for  the  discount ; 

As  the  amount  of  100/.  or  dollars,  at  the  rate  and 
time  given,  is  to  the  interest  of  the  same  sum  at  the 
same  rate  and  time,  so  is  the  given  sum  to  the  discount 
required. 


I 


DISCOUNT.  207 

Kx.  1.  What  is  the  present  worth  and  discount  of 
620  dollars,  due  4  years  hence  at  6  per  cent,  per  annum 
discount  ? 
6 

4 

____  • 

24    Interest  of  S 100  at  6  per  cent,  for  4  years. 

100 

124  Amount  of  glOO  for  4  years  at  6  per  cent, 

124  ;  100  ::  620 

100 
g 

124)62000(600 
620 


00 
g620 
500  present  worth. 

§120  discount. 

Proof.  or  thus  ; 

500  124  ;  24  :  :  620 

6  24 


30.00  2480 

*  1240 

120.00  124)14880(120 

500  124 

2620  248 
248 

620 

120  discount. 

500  present  worth. 


20S  DISCOUNT. 

Ex.  2.   What  is  the  discount  of  g7 18.75  for  5  years 
at  5  per  cent,  per  annum  ?         Ans,  143  dols.  75  cts. 

Ex.  3.  What  is  the  present  worth  of  1092^.   13s.  due 
5  years  henc6  at  6  per  cent,  per  annum  ? 

Answer,   846/.  10«. 
Ex.  4.  What  is  the  present  worth  of  284  dols.  28  cts. 
due  8  months  hence  at  4V  per  cent,  per  annum  ? 

Answer,  ii76  dollars.    . 

Ex.  5.  What  is  the  discount  of  250/.  10s.  6d.  due  2 

years  and  4  months   hence,  at  6^^  per  cent  per  annum  ? 

Answer,  31/.   17s.  8|:/. 
Ex.  6.  What  is  the  present   w^orth  of  1000/.  due  3 
years  and  7  months  hence,  at  5|  per  cent,  per  annum  ? 
Answer,   8S9/.  3s.  2d.  /j\% 
Ex.  7.  What  is  the  present  worth  of  G40/.   l0>.  due 
10  years  and  2  months  hence,  at  4^   per  cent,  per  an- 
num discount  ?  Answer,  43Q/.  9s.  0^/.  i|| 

Ex.  8.  What  is  the  discount  of  740  dols.  30  cts.  due 
7^  years  hence,  at  6j  per  cent,  per  annum  ? 

Answer,  242  d«ls.  68  cts.  -/-j^g- 
Ex.  9.  What  is  the  present  worth  of  500  dollars,  one 
half  payable  in  6  months,  and  the  other  half  in  8  months, 
discount  at  6  per  cent,  per  annum  ? 

Answer,  483  dols.   10  cts.  -^\W 
Ex.  lO.  AVhat  diflference  is  there  between  the  interest 
of  600  dollars  for  1  year  and  9  months  at  6  per  cent,  per 
annum,  and  the  discount  of  the  same  sum  at  the  same 
rate  and  for  the  sametime  ?  Ans.    5  dols.  98  cts.  |^|f 


Discount  in  business  is  generally  reckoned  in  the  same  man- 
ner a»  common  interest.  .  1      j-r 

When  the  sum  is  not  very  large,  and  the  time  short,  the  dit- 
ference  between  the  discount  and  the  interest  is  a  mere  trifle  ; 
but  when  the  sum  is  large  and  the  time  considerable,  their  dif- 
ference then  becomes  essential,  and  the  sum  should  be  calculat- 
ed on  correct  discount  principles. 


(209  ) 


PROFIT  AND  LOSS 

Is  a  rule  that  discovers  what  is  gained  or  lost  on  thifc 
prime  cost  in  the  purchase  and  sale  of  goods,  and  it  tea- 
ches how  to  to  fix  the  price  of  their  goods  so  as  to  gain 
so  much  per  cent. 

Questions  in  this  rule  are  performed  bj  the  Rule  of 
Three  Direct,  upon  this  principle,  that  quantities,  or  sums 
of  money,  which  gain  or  lose  at  the  same  rate,  are  to 
one  another  as  their  gains  or  losses. 

Ex.  1.  A  tallow  chandler  has  this  day  purchased  mot- 
tled soap,  at  102s.  6d.  per  cwt.,at  how  much  per  lb.  must 
he  retail  it  out  to  gain  10  per  cent,  profit  ? 

L.  s.     d. 

:     110     ::       102   6 -i- 112 

102     6 

2000  2£0 

1 100      .- 
5^ 


2.000)11.275 


5.6375 


L.  5.6315  and =ls.^=:is.  0|rf.  nearly, 

112 

Ex.  2.  How  much  per  cent,  is  gained  at  the  rate  of  2d. 
in  a  shilling  ?  Answer  16Z  13b.  4rf. 

Ex.  3.  If  3  dollars  he  gained  in  selling  at  21  dollars,  at 
what  rate  per  cent  is  that  ?  Answer  I6f  per  cer* 

18* 


£10  PAUTKERSHIP. 

Ex.  4.  Three  pounds  of  tobacco  are  bought'at  5*.  9(1. 
and  sold  for  7s.  6d.,  what  is  the  gain  upon  the  sale  of  what 
cost  loo/.  Answer  30/.  8s.  S^c?. 

Ex.  5.  Bought  cheese  at  3/.  3s.  per  cut.,  and  sold  it 
again  at  lO^d.  per  lb. :  what  is  the  gain  per  cwt.  suppos- 
ing the  loss  in  weight  to  be  4lb.  per  cwt. 

Answer,  L.  1    11   6  gain  per  cwt. 

Ex,  6.  Bought  silk  stockin^^s  at  ^4  25  per  pair,  what 
must  thej  be  sold  for  to  gain  20  per  cent  profit  ? 

Answer,  gS.lO. 

Est:  r.  If  375  yards  of  cloth  be  sold  for  290i.  and  there 
be  20  per  cent,  profit,  what  did  it  cost  per  yard  ? 

Answer,  12s.  lOfo?. 

Ex.  8.  If90  English  Ells  of  Cambric  cost  120  dolls, 
for  how  much  must  1  sell  it  per  yard  to  gain  18  per  cent  ? 

Answer,  gl  25i|. 

Ex.  9.  A  plumber  sold  5  fother  of  lead,  for  \02l2s.6d. 
(the  fother  being  19|  cwt.),  and  gained  after  the  rate  of 
12/.  10s.  percent.  :  what  did  it  cost  him  per  cwt.  ? 

Ans-vver,  l8s.  7^^ 

Ex.  10.  Bought  218  yards  of  cloth,  at  the  rate  of  8s.  6d. 
per  yard,  and  sold  it  for  lOs.  ^d.  per  yard  :  what  was  the 
gain  of  the  whole  ^  Answer,  19/.  19s.  8d. 

Ex.  1 1.  Paid  69/.  for  one  ton  of  steel,  which  is  retailed 
at  Sd,  per  lb.,  what  is  the  profit  or  loss  by  the  sale  of  12 
tons?  Answer  X.  68 gain. 


PARTNERSHIP 

Is  a  general  rule,  by  which  merchants,  &c.,  trading  in 
company  with  a  joint  stock,  are  enabled  to  ascertain  each 
person's  particular  share  of  the  gain  or  loss,  in  proportion 
to  his  share  in  the  stock. 

This  rule  divides  itself  into   two  parts,  vi«.   1.  Part- 
-irship  without  regard  to  time  :  and  2.  Partnership  with 
me. 


PARTNEKSHIP.  2U 

I.  Partnership  without  Time. 

Rule,  "  As  the  whole  stock  is  to  the  whole  gain  or 
loss,  so  is  each  man's  share  in  the  stock  to  his  share  of 
the  gain  or  loss." 

Ex.  1.  Two   merchants  embark  in  business,  the  one 
puts  in  as  capital  X.5550,  a  id  the  other  L.34i20,  and  they 
gain  in  the  first  year  i.l 260,  what  is  each  man's  gain  ? 
L.5550 
3420 

8970  —joint  stock. 

8970/.  :  1260/.   :  :  5550/.  :  779/.  12s.  nearly  ;    of  course 
the  profits  of  the  other  are  1260/— 779/.  12s.=480/.  8s. 

Ex.  2.  Three  persons  trade  together  :  A  puts  in  li/0/. ; 
B  150.  ;  C  200/.  ;  and  they  gain  900/.:  what  is  each 
man's  gam  ?  Ans.  A  200--B  300 — C  400. 

Ex.  3.  A,  B,  and  C,  enter  into  partnership  ;  A  puts  in 
S640/.,  B  4S20/.,  and  C  5000/.,  and  they  gained  8670/.; 
what  is  each  man's  share  in  proportion  to  his  stock  P 

Ans.  A  2344/.  13s.  nearly — B  3104/.  i  4s. — C  3220/.l3s. 

Ex.  4.  Four  merchants.  B,C,  D,  and  E,  nmke  a  stock  ; 
B  put  in  2270/.,  C  3490.,  D  11 50/.  and  E  4390  ;  in  trad- 
ing they  gained  4280/.  I  demand  each  merchant's  share 
ot  the  gain  ?  Ans.  JB  859/.  16s.  nearlv—C  1321/.  175. 
6d D  435/.  lis.  6d,  nearly— E  1662/.  \5s. 

Ex    5.  Three  persons,  D,  E,  and  F,  join  in  company  ; 
D's  stock   was   3750/.,  E's  2800/.,  .md   F's  250O/.,  and 
at  the  end  of  12  months  they  gained  3420/. ;  what  is  each 
man's  particular  share  ot  the  gain  ? 
Ans.  D  1417/.  2s.  6^fl/.-E  1058/.  2s.  5c/.— F  944/.  15s.  |d!. 

J/.  Partnership  with  Time. 

Rum.  As  the  sum  of  the  product  of  each  man's  mo- 
ney i.nd  time  is  to  the  whole  gain  or  loss,  so  is  each 
man's  product  to  the  share  of  the  gain  and  loss. 


21S  PARTNERSHIP  WITH    TIME. 

Ex.  1.  Two  persons  lay  out  1500^.  in  trade,  m  the 
proportion  of  3  to  2  :  that  is,  A  put  in  900L,  and  B  600l. ; 
A  leaves  his  money  in  the  concern,  9  months,  and  B 
docs  not  want  his  for  12  months  :  what  profits  belong  to 
each,  supposing  they  gain  250^  ? 

X..900  X     9  =  8100 
600  X    12   =  7200 


15300 
15300     :   250  : :     8100 
250 


153.00)20250.00(132^.  7s. 
Ans.  A*s  share  of  profit  L.  1 32    7  0 
B's     -    -     -    -         lir  13  0 


Z<.250    0  0 


Ex.  3.  A  puts  into  a  concern  208O/.  for  2  months,  B 
970/.  for  5  months,  and  C  400^  for  15  months  ;  they  gain 
among  them  650/.  ;  what  must  each  receive  for  his  share 
of  profit  ? 

Ans.  180/.  3s.  A*s  profit  nealy. ;  210/.  B's  profit; 
259/.  17  s.  C's  profit.^ 

Ex.  3.  Three  merchants  join  in  company  for  18 
months  :  D  put  in  500/.  and  at  5  months'  end  took  out 
200/. ;  at  10  months'  end  put  in  500/.,  and  at  the  end  of 
14  months  takes  out  130/. ;  E  puts  in  400/,  and  at  the 
end  of  3  months  270/.  more  ;  at  9  months  he  takes  out 
140/.,  but  puts  in  100/.  at  the  end  of  12  months,  and 
withdraws  99/.,  at  the  end  of  15  months.  F  put  in  900/., 
and  at  6  months  took  out  200/.;  at  the  end  of  11  months 
puts  in  600/.,  but  takes  out  that  and  100/.  more  at  the 
end  of  13  months.  They  gained  200/.  I  desire  to 
know  each  man's  share  of  the  gain  ? 

Ans.  57/.  nearly  D's  gun ;  59/.  7s.  5d,  E's  gain  } 
83/.  12s,  Id,  =  F's  gain. 


(  213  ) 


ALLIGATION 


Teaches  to  mix  things  of  different  values,  so  as  to  as- 
certain the  price  of  the  mixture.  There  are  two  cases  in 
this  rule. 

T.  To  find  the  mean  value   of  a   mixture  composed  of 
several  quantities  of  ditierent  values. 

Rule.  Multiply  each  quantity  by  its  respective 
value,  and  divide  the  sum  of  the  products  by  the  sum  of 
the  quantities. 

Ex.   1.  A  tea-dealer  mixes  3i  cwt.  of  tea,  at  9s.  per 
lb.,  with  2  cwt.,  at  7s.  and  4-i  cwt  at  5s.   6d.,  at  how 
much  per  lb.  can  he  sell  the  whole  mixture  ? 
3^X112  =  392")  r392x9     ==3528 

2     X  112  =  224.  land-}  224  X  7     =1568 
4J  X  1 12  =  4-76  J  i4>7Q  X  5i  =  2618 

1092  )lTU{7s.0^d,f^ 

7644 


Answer  -  7s.  O^d,  .  .70 

Ex.  2.  What  is  a  lb.  of  sugar  worth  which  is  com- 
pounded of  3  cwt.  at  46s. :  2  cwt.  at  59s.  ;  l|  cwt.  at 
84^. ;  and  56#.  at  60s.  ?  Answer,  6ifl?. -fg-f. 

Ex.  3.  What  is  the  average  earnings  of  workmen,  4  of 
whom  earn  10  dollars  each  per  week  ;  8  earn  9  dollars 
each  ;  and  12  will  get  only  6  dolls.  50  cts.  each  ? 

Answer,  7  dolls.  91f  cts. 

Ex.  4.  A  tobacconist  mixes  80  lb.  of  tobacco  at  SOJ. 
per  lb. ;  150  lb.  at  2s.  3d.  per  lb.  ;  and  40  lb.  at  3s.  lOd. 
per  lb. ;  wiiat  will  be  the  value  of  the  mixture  per  oz.  ? 

Answer,  I|d.  nearly. 


214  ALLIGATION. 

11.  To  find  how  much  of  difterent  things  of  different 
values,  must  be  taken,  in  order  to  make  a  mixture  of  a 
certain  mean  value. 

Rule  (1).  Set  down  the  names  of  the  things  to  be 
mixed,  together  with  their  prices;  then,  finding  the  dif- 
ference between  each  of  these,  and  the  proposed  price  of 
the  mixture ;  place  these  differences  in  an  alternate  or- 
der, and  they  will  shew  the  proportion  of  the  ingredients. 

Ex.  1.  Orange  wine,  at  9s.  per  gallon,  is  to  be  mixed 
with  raisin  wine  at  6s.  per  gallon ;  what  will  be  the  pro- 
portions, so  as  to  sell  the  mixture  at  7s,  per  gallon  ? 

Proposed 
Orange  -  9s,  C  price,  r  1  "J  A  mixture  therefore  of  these 
J     7  s.   J     f  wines  in  the  proportion  of  one 

j  J     r  orange  to  two  raisin,   will  be 

Raisin    -  (is.'  (_  2)  the  answer. 

Ex.  2.  A  spirit  at  10  shillings,  and  another  at  12  shil- 
lings per  gallon,  are  to  be  mixed  with  low  wines  at  6  and 
5  shillings,  in  order  to  produce  a  mixture  worth  9  shil- 
lings per  gallon  ;  what  must  the  quantities  of  each  be  ? 

3  Theansweris,  3  gallons  at  16s., 

4  4  at  1 2s. :  7  at  6s. ;  and  3  at  5s. ; 
7  will  make  a  mixture  that  may  be 
3  sold  for  9  shillings  per  gallon :  for 


Spirit,  -       16 

Ditto,    -  ^1 
Wine,  - 

Ditto,    -  ^ 


^ 


3  X  16  =  48 

4  X  12  =48 
7  X  6  =  42 
3  X     5  =  15 

—  • — —  T53 

17  I53and-— =9s,    Pro«f, 

17 

Ex.  3.  A  tea-dealer  would  mix  four  sorts  of  tea  toge- 
ther, viz.  at  4s.,  4s.  6d.,  5s.  6rf.,  6s.,  and  7s.  per  lb. ;  in 
order  that  he  maj  sell  the  whole  mixture  at  5s.  6rf.  per 
lb.,  what  proportion  of  each  will  he  use  ? 

Ans.  lA  lb.  at  4s. ;  |  lb.  at  4s.  6d.  1  lb.  at  6s, ;  and 
1 J  lb,  at  7s,  ;  and  as  much  as  you  please  at  5s.  6d. 


^ 

at  48 

X 

—  42 

1 

—  27 

H 

—  24 

lb. 

c^s. 

1 

at  48 

1 

—  42 

3 

—  27 

2 

—  24* 

ALLIGATION.  215 

Ex.  4.  How  much  cofifee  at  48  cts.,  42  cts.,  27  cts. 
and  24  cts.  per  lb.  will  compose  a  mixture  worth  30  cts. 
per  lb. 

lb.     cts.  lb.      cts. 

Ans.     6  at  48 

3  —  42     or 
12  —  27 
24  —  18 

OR    THUS. 

lb.      cts. 

Ans.     1  at  48 

2  —  42 

6—27 

4  —  24 

III.  When  the  prices  of  all  the  things  to  be  mixed 
are  given,  likewise  where  the  quantity  of  one,  and  the 
mean  rate  are  also  given,  to  find  the  several  quantities 
of  the  others. 

Rule.  (1).  Take  the  difference  between  each  price 
and  the  mean  rate  as  before.  (2).  As  the  difference  of 
that  thing,  whose  quantity  is  given,  is  to  the  rest  of  the 
ditferences  severally  ;  so  is  the  quantity  given  to  the 
several  quantities  required. 

Ex.  1.  A  rectifier  of  compounds  has  200  gallons  of 
spirit  that  he  can  sell  for  12s.  6d,  per  gallon,  but  he 
means  to  mix  it  with  three  other  kinds  of  spirit  at  1 3s.  4c?., 
at  I5s.,  and  18s.  4i.,  per  gallon,  in  order  that  he  may 
sell  the  whole  at  14s.  2i.  per  gallon  ;  how  much  must 
he  use  of  each  ? 

I  reduce  the  several  prices  to  pence,  which  stand  as 
follows  : 

170 


150--.  50 

160.1  10  50;  10:  :  200  ;  40 

180^  10  50  :  10  :  ;  200:  40 

220—'  20  50  :  20  :  :  200  :  80 


*  Note.— A  variety  of  answers  can  be  obtained  to  these  ques- 
tions,  by  linking  tliem  different  ways,  they  may  also  be  made 
infinite  by  multiplying  or  dividing  any  result  by  one  common 
number. 


40  at  13  4  = 

26 

40  at  15  0  = 

30 

80  at  18  4  = 

73 

216  ALLIGATION. 

The  answer  is  ;  to  200  gallons,  at  12s.  6d.,  must  be 
added  40  at  l3s.  4d.,  40  at  l5s.,  and  80  at  18s.  Ad. ;  the 
truth  of  which  is  proved  thus  ; 
200  at  1:2     6  =   125     0  0 
13  4 
0  0 
6  8 

360  255     0  0  and =  14i,.  2d.  Proof. 

360 

Ex.  2.  A  grocer  has  100  lb.  of  tea  worth  4s.  per  lb. 
which  he  means  to  mix  with  others  at  12s.  3d.,  10s.,  and 
6ii.  per  lb.  ;  in  order  to  sell  the  whole  at  8s.  how  much 
of  each  must  be  used  ? 

Ans.  100  lb.  at  4s. ;  100  lb.  ^t  12s.  3d. ;  200  lb.  at  10s. ; 
and  212^  lb.  at  6s. 

IV.  When  the  price  of  each  thing  is  given,  also  the 
quantity  and  the  mean  rate,  to  find  how  much  of  each 
sort  will  make  that  quantity. 

Rule.  (1).  Take   the  difference  between   each  price 
and  the  mean  rate  as  before  :  then  (2).  As  the   sum   of 
the  diflferences  is  to  each  particular  difference,  so  is  the 
quantity  given  to  the  quantity  required. 

Ex.  1.  A  wine  merchant  means  to  mix  860  gallons  of 
wine  to  sell  for  8s.  a  gallon,  out  of  other  wines  that  he 
already  sells  for  12s.,  9s.  6s.,  and  5s.  per  gallon,  how 
much  must  he  take  of  each  ?  .    . 


8 


1  12-^       3  10:  3  ::  860:  258 

2         10  :2  ::860:  172 

10:  J  ::860:    8S 

10:4;:  860  :  344 


2-         6 
9  2 

6^  1 


Sum  of  differences  =  <0 

The  answer  is  258  gallons  at  1 2s.  ;  172  at  9s. ;  86  at 
6s. }  and  344  ai  5s.  per  gallon,  /iiay  be  mixed  and  sold 
at  8s.  per  gallon. 


rosiTioN".  217 

Ex.  2.  A  ^oKlsmit'n  has  (our  sorts  of  goh],viz.  ofS4, 
10,  IS,  and  15  carats  fine,  vvis!ies~  125  oz.  of  the.  fineness 
of  17  carats,  how  much  will  he  want  of  each  sort  ? 

Ans.  14.  02.  16  dwt.  1 1 /^  e;r.  of  24.. ;  7  oz.  8  dwt.  5j-f 
gr.  oflO.  ;5l  oz.  17  dwt.  15^\  gv.  oi  IS. ',  51  oz.  17  dwC 
l5,Vgr.ofl5.? 

Ex.  3.  A  drug  grinder  has  hark  worth  l6.s.  per  lb., 
some  at  lOs.,  and  some  at  4s. ;  but  he  is  desirous  of 
making  up  two  parcels,  viz.  one  containing  a  cwt.  at  9s., 
a«d  the  other  84- lb.  at  12s.;  what  proportions  of  each 
must  be  used  ? 

Ans.  43J.J  lb.  at  16s.  ;  8-^^^  lb.  at  10s.  ;  60^\  lb.  at  4s. 
for  112  lb.  5  48  lb. ;  12  lb.  ;  24  ib.  for  84  lb.  ? 


POSITION 


Position,  or  as  it  is  sometimes  called,  the  Rule  of 
False,  is  a  rule,  that  by  means  of  any  supposed  num- 
bers, others  thart  are  true,  and  that  answer  to  the  terms 
of  the  question,  are  found.  There  are  two  kinds  of  Po- 
sition, viz.  Single  and  Double. 

Single  Position  is  performed,  by  using  a  supposed 
number,  and  working  with  it  as  the  true  one,  till  the 
real  number  is  found. 

Rule.  Take  any  number  and  perform  the  work  with 
it,  as  if  it  were  the  right  number  :  then  say,  As  the  re- 
sult of  this  work  is  to  the  position,  so  is  the  result  in  the 
question  to  tlie  number  required. 

Ex.  1.  A  person  counting  some  guineas,  being  asked 
how  many  he  had,  repli-ed  :  "  If  you  had  as  many,  and 
as  many  more,  and  half  as  many,  and  one  quarter  as 
many,  you  would  have  26  k"  How  many  had  the  per- 
son who  was  counting  his  gold  ? 

19 


SIS- 


POSITION. 


By  way  of  supposition,  I  take  80  as  the  number;  then, 
by  the  terms  of  the  question,  it  will  be 

80  96 

As  many  more,   80     220  :  264  :  :  80  96 

Half  as  "many,     40  80  48 

ith.  as  many,     20 24 

220)211 20(96  Ans. 

220 264  Proof. 

Ex.  2.  A  person  after  spending  A,  ^,  and  -},  of  his 
money,  finds  he  had  500/.  left,  wiiat  was  his  original 
propel  ty  ? 

I  take  a  number  divisible  by  2,  4,  and  6,  for  the  sup- 
position, viz.  60. 

Suppose    60     60  —  55  ZI 5,  tiierefore         Proof. 
—     As  5  :  60  :  :  300         ^  ==  3000 
4-  30  60       A  =  1500 


15  

10  5)30.000 


55O0 


55        Answer,  L.6.O0O  500  rem. 

Ex.    3.   Three   persons  bought  goods   at   Baltimore, 

which  cost  600  dollars.     The  first  person  was  to  have  a 

third  part   more  than  the  second,  and  the  third  a  fourth 

part  nK)re  than  the  first ;  what  was  each  man's  share  .? 

Ans.  g200  first  person's  share,  Si 50  second  share, 

.  g250  third  share. 

Ex.  4.  In  a  leaky  vessel  there  were  three  pumps  of 
ilitferent  capacities  ;  the  first  would  enipty  the  hold  of 
the  ship  in  20  minutes,  the  second  vvould  require  double 
that  time,  and  the  third  would  not  perform  the  business 
in  less  than  an  hour ;  how  long  would  all  three  together 
take  in  doing  it  ?  Answer,  1 1  minutes  nearly. 


(  '^19  ) 


DOUBLE  POSITION 


QuKSTiONS  in  this  rule  are  resolved  by  rnakino:supp<>- 
sitio?is  of  two  nusiibers,  whicli  maif  both  prove  false  ;  ill 
that  case  the  errors  are  made  to  correct  each  other. 

Rule.  (1.)  Place  each  error  against  its  respective  po- 
sition, and  multiply  them  cross  ways.  (2.)  If  the  errors 
are  alike,  that  is,  both  greater  or  both  less  than  the  given 
number,  take  their  difference  for  a  divisor,  and  the  dif- 
ference of  their  products  for  a  dividend.  But  if  unlike, 
take  their  sum  for  a  divisor,  and  the  sum  of  their  pro- 
ducts for  a  dividend,  the  quotient  will  l)e  the  answer. 

Ex.  1.  Three  persons  have  obtained  the  2O,000Z.  prize 
in  the  lottery,  and  it  is  to  be  so  divided,  that  the  second 
is  to  have  600^.  more  than  the  first,  and  the  third  800/. 
morg  than  the  second,  what  is  each  person's  share  ? 
Suppose  the  first  had  5000  Suppose  the  first  had  5600 
Then  the  second  had    5600  The  second  had     0200- 

ajid  the  third  had    6400  The  third  had        7O0O 


27000  too  little  by  300O        1 8800 
[too  little  by  1200. 
3000     5000")  fSOOO   X  5600  ZZ  16800000 

X  Vthat  is,-< 

1200     5600j  (1200   X   50OO  n  GOOOOOO 

Difference  of  Products         -        lObOOOOO  = 

[dividend. 
3000  ^  1200  =  18(0   (diff  of  errors)  for  a  divisor. 
10.800.{)00  GOOO 

Therefore, . —  =  L.COOO  dtiOO 

1800  7400 

L.  20.000  Proof 


220  COMPOUND    INTEREST    AN1>    ANNUITIES. 

Ex.  1.  A  genfleman  at  Christmas,  wished  to  give  se- 
veral poor  families  5  shillings  each,  but  he  found  he  had 
16s.  8(7.  too  little;  he  then  gave  them  3s.  6c?.  each, 
and  found  he  had  4s.  4i.  left,  how  many  families  we-re 
there?  Answer/ 14  families. 

Ex.  3.  A  person  purchased  a  house  and  land,  togeth- 
er with  a  carriage  and  horses,  for  13  000  dollars;  he 
paid  4  times  the  price  of  the  carriage  and  horses  for  the 
land,  and  3  times  the  price  of  the  land  for  the  house, 
Vhatwas  the  value  of  each  separately  ? 

Ans.   gCOO  carriage  and  horses,  825400  land, 
g  12,000  house. 


COMPOUND  INTE^^EST  AND  ANNUITIES. 


Compound  Interest,  or  interest  «pon  interest,  is 
that  which  is  paid  not  only  for  the  use  of  the  money 
lent,  but  also  for  the  use  of  the  interest  as  it  becomci:  due. 

There  are  two  methods  of  working  Problems  in  this. 
Rule,  viz.  by  Common  Arithmetic;  and  by  Decimals  j 
1  shall  give  examples  under  each. 

1.  liy  Common  Arithmetic, 

Rule.  1.  Find  the  amount  of  the  given  principal  for 
the  time  of  the  first  payment  by  simple  interest.  (2.) 
Consider  this  amount  as  the  principal  for  the  second  pay- 
ment, the  amount  of  which  is  to  be  calculated  as  before, 
and  so  on  through  all  the  payments  to  the  last,  stdl 
reckotiing  the  last  amount  as  the  principal  for  the  next 
payment. 


COMPOUND    INTERES-n  221 

Ex.  1.  What  is  the  amount  of  550L  for  three  years, 
at  5  per  cent,  compound  interest  ? 

20)550     0     0  iiven  principal. 

27  10     0  first  year's  interest. 

20)577   10     0  second  year's  principal. 

28  17     0  second  pear's  interest. 

20)606     7     6     third  year's  principal 
30     6     4-|  third  j'ear's  i..terest. 

Answer    -     63(>  13  10| 

V.x,  2.  What  is  the  amount  of  400^  for  four  years, 
at  5  per  cent,  compound  interest  ?    Ans.  48(1^  4s.  O^d, 

Ex.  3.  What  is  the  compound  interest  of  COO  dols. 
for  5  years  at  5  per  cent  per  annum  .^  Ans.  Si  65.7689375 

II.  By  Decimals, 

Rule.  1.  Find  the  amount  of  \L  for  a  year,  at  the 
given  rate  per  cent.  2.  involve  the  amount  thus  found, 
to  such  a  power  as  is  denoted  by  the  nun»ber  of  years, 
o.  Multiply  this  power  by  the  pri;icipai  or  i^iven  sum, 
and  the  product  Mi!!  be  the  amount  required.  4.  Sub- 
tract the  princip.il  from  the  amount,  and  tlie  remainder 
will  be  the  interest. 

Ex.  1.  What  is  tite  compound  interest  of  550l,  for  3 
years,  at  5  per  cent,  per  annum  ? 

1.05  =  amount  of  l7.  for  a  year,  at  5  per  cent. ; 
Then  1.05  X  1.05  X  1.05  =  1.157625,  and 
1.157025  X  550  ==  f)35  69375  ^  amount^ 
636.69375—550  =  86.693:5  —  86/.   ISs.   lO-J-f^. 

Ex.  3.  Wliat  is  the  amount  of  ! 00  dols.  for  4  yearSj 
at  6  per  cent,  per  annum,  compound  interest  ? 

Answer,  gl26  247696. 

Ex.  S.  AVhat  is  the  compound  interest  of  620/.  for  6 
years,  at  5  per  cent...^  Answer,  171L  58.  lOd^ 


222 


COMPOUND    INTEREST    AND    ANNUITIES. 


A  TABLE. 

Shewing  the  Sum  to  which  J^  or  gl  Principal  will  in- 
crease at  5  per  cent.  Compound  Interest,  in  any  num- 
ber of  years  not  exceeding  a  hundred. 


Yrs 

Amount, 

Yrs. 
26 

Amount. 

Yrs. 
51 

Amount. 

Yrs. 

Amount, 

1 

1.05 

3.555572 

12.040769 

76 

40.774320 

2 

I  10:5 

27 

3.733456 

52 

12.642808 

77 

4-813036 

3 

1.157625 

28 

3.920129 

53 

13.274948 

78 

44953688 

4 

1.215506 

29 

4  11 6135 

54 

13.938696 

79 

47.201372 

5 

1.276^281 

30 

4.321942 

55 

14  635630 

80 

49.561441 

6 

1.340095 

31 

4.53803iy 

06 

15.367412 

81 

.52  039513 

7 

1.407100 

32 

4  764941 

57 

16.135783 

82 

54.641488 

8 

I  477455 

33 

5.U031S8 

58 

16.942572 

83 

S7:373563 

9 

1.551328 

34 

5.253347 

59 

17.789700 

84 

60  242241 

10 

1.62^894 

35 

5.516013 

60 

18  079185 

85 

63.254353 

11 

1.710339 

•36 

5.791816 

61 

19.613143 

86 

66.417071 

12 

1.795856 

37 

6  081 406 

62 

JO  593802 

87 

69  737924 

13 

1.885649 

38 

6.3S5i77 

63 

21.623492 

88 

73.224820 

14 

1.979931 

39 

6.704751 

64 

22  704667 

89 

76  886061 

15 

2.0789?8 

40 

7.03998^ 

65 

23.839900 

90 

80.730365 

16 

2.182874 

41 

7.391988 

66 

25.031395 

91 

84  766883 

17 

2.292018 

42 

7.761587 

67 

26.283190 

92 

89.005227 

18 

2.406619 

43 

8.149666 

6S 

27.597C-64 

93 

9.5  455438 

19 

2.526950 

44 

8..'}  57 150 

69 

28.97754^ 

94 

98.128268 

2U 

2-623297 

45 

8.985037 

70 

:0.425425 

95 

103.034676 

21 

2.785962 

46 

9  434258 

71 

.1.947746 

96 

» 08- 1864 10 

22 

2.925260 

47 

9  9u597] 

72 

33  545134 

97 

11 3. 59.0730 

23 

3.071523 

48 

10.4012g1 

73 

35.222390 

98 

119.275517 

24 

3.225099 

49 

10.92  i3.>;i 

74 

36  983510 

99 

125.239293 

25 '3386354 

50 

11.467399 

75 

^8.832685 

100 

131.5012.57 

1.     To  find  by   means   oi'  tlie  table  what  any   sum  will 
amount  to  in  a  given  number  of  years. 

RuLB.  Multiply  the  number  in  the  table,  opposite 
to  the  term  of  years,  by  the  sum,  and  the  product  will 
be  the  answer. 

Ex.  1-  To  what  sum  will  500/.  amount  to  in  44 
years,  at  5  per  cent,  compound  interest  ? 

Opposite  to  44.  in  the  table  I  find  8.5.07 150,  this  I  mul- 
tiply by  500,  and  the  answer  is  4278/.  11*-.  6(i. 


COMPOUND    INTEREST    AND    ANNUITIES.  22o 

Ex.  2.  What  will  350^  araount  to  in  25  years,  at  5 
per  cent,  compound  interest  ? 

Answer,   1185/.  4s.  5^d.  nearly. 

Ex.  3.  A  prudent  young  man  marries  at  the  ajre  of 
22  ;  the  fortune  which  he  has  with  his  wife  is  2500/., 
half  of  which  he  readily  gives  into  the  hands  of  trus- 
tees to  be  accumulated  at  5  per  cent  compound  inte- 
rest;  what  will  it  araount  to,  supposing  he  lives  32 
years,  which  he  may  reasonably  expect  ? 

Answer,  5956/.  3s.  6^d, 

Ex.  4.  The  year  1808,  is  tliat  in  which  the  late  Mr. 
Pitt  calculated  there  would  be  four  millions  surplus  to 
be  applied  to  the  payment  of  the  national  dehi  of  En- 
gland :  1  demand  how  much  this  single  four  millions 
will  accumulate  in  half  a  centurj',  at  5  per  cent,  com- 
pound interest  ?  Answer,  45,8oQ,596/- 
(See  other  questions  on  this  subject  after  the  next  table.) 

II.  To  find  the  number  of  years  in  which  a  given  sum 
will  increase  to  another  given  sum,  in  consequence 
of  being  improved  at  Compound  Interest. 

Rule.  Divide  the  latter  sum  by  the  former,  and 
the  sum  in  the  table  which  is  nearest  to  the  quotient 
will  shew  the  terms  required. 

Ex.  1.  In  what  time  will  200?.  increase  to  1500/.,  if 
improved  at  5  per  cent,  compound  interest  ? 
1500 

7.5.      The  nearest  number  in   table  I.   to  7.5  is 

2fO 

7.391988,  opposite  to  which  is  41,  the  number  of  years. 
Of  cuuise  2(i0/.  in  a  little  more  than  forty-one  years 
wouhl,  by  beini;  accumulated  at  compound  interest,  at  5 
per  cent.,  amount  to  1500/. 

Ex.  2.  In  what  time  will  100/.  increase  to  600/ 
sanie  rate  of  interest  ^  Ans.  33  year^ 

Ex.  2.  In  what  time  will  8G0/.  increase  to  K 
Ans.  between  50  an 


,^24 


COMPOUND  INTEREST  AND    ANNUIIIES 


Ex.  4.  In  how  long  would  five  millions  be  in  pajino;  ti.e 
national  debt,  which  m  January,  1806,  was  upwards  of 
580  millions  ?  Ans.  between  97  and  98  years. 

Ex.  5.  Admiral  Rainier  left,  in  1808,  25,000/.  to- 
wards paying  off  the  national  d(^bt,  \.  hen  will  if  have 
accumulated  to  a  million  at  5  per  cent  compound  in- 
terest i  Ans.  76  years,  nearly. 

TABLE  II. 


Shewing  the  sum  to  winch  1/  per  annum  will  increase  at 
5  per  cent.  Compound  Interest,  in  any  number  of  years 
not  exceeding  a  hundred. 


YrsT 

Amount. 

Yib 

Amount 

Yis.  1  Vinouut. 

Yrs. 
76 

Amount. 

X 

1,0000 

26 

51,1135 

51 

220,8154 

795,4864 

2 

2,0500 

27 

54,6691 

52 

232,8562 

77 

836,2607 

3 

J,l>25 

28 

58,4026 

53 

245.4990!  78 

879,0738 

4 

4,U01 

29 

6  \  1227 

^4 

r58,7739,  79 

9-^4,0274 

.5 

5, 5256 

iO 

66,4388 

55 

272,7126 

m 

971,2288 

6 

6,8019 

31 

70,7608 

56 

287,318 

81 

1020,790''i 

7 

8.1420 

''o 

7.S^^'98b 

57 

302,7j57 

82 

1072,.-.293 

8 

9,5491- 

oo 

8','.u6.iB 

.^8 

318,851^ 

83 

1127,4713 

9 

11,0  66 

34 

85,  u  670 

59 

335,  .■940 

84 

1184,8448 

10 

12,577^ 

;.-; 

90,3203 

60 

353,583;'  85 

1245,0871 

11 

'.4,-:?068 

jG 

95,8363 

61 

372,2629  86 

1308,34J4 

12 

15,9171 

3? 

101  ^281 

62 

391,8760'  87 

1374,7585 

13 

17,7130 

;8 

;  07,7095 

63 

412,4698  88 

1441, 4964 » 

14 

19,5986 

39 

114,095' 

64 

434,0933  89 

1517,7212 

15 

21.3786 

40 

120,7998 

65 

456,7980  9(' 

1 594,6073 

16 

23,6575 

41 

I27.83^h 

66  480.6379 

91 

\675,'i377 

17 

25,,-'40l  \42 

135,2317 

67 

505,6698 

92 

1760,1045 

18 

:8,1328  43 

142,993; 

68 

5.1  9583 

93 

1849.1098 

19 

30,53^/0  44 

r  1,14-^0 

69 

559,5510 

94 

1942,5653 

20 

33,06^9 

45 

1 5-^,7(02 

70 

588.5285  95 

2040,6935 

^1 

33,7192 

4f> 

1-38.685^ 

71 

618,9549  96 

2143,7282 

22 

38,505S 

47 

178,119'", 

72 

6'-.0,902r| 

97 

2251,9146 

23 

41.430.' 

.18 

188,0254 

73 

684,4478 

98 

2365,6x03 

24 

44,5020 

49 

19.^\4267 

74 

719,6702 

99 

2484,7859 

2.. 

4.  .7271 

-0  2US.34.S  1 

7.5 

756,6h37 

100 

2610,0232 

COMPOUND  INTEREST  AND  ANNUITIES.  2S^5 

I.    To  find  in  what  time  a  given  annuity  will  amount  te 
a  given  sum  at  compound  niterest. 

Rule.  Divide  the  given  sum  hy  the  given  annuity, 
and  the  number  in  the  table  nearest  to  the  quotient  will 
be  the  answer. 

Ex.   1.  A  person  owes   lOOO/.  and  resolves  to  appro- 
priate 201.  per  annum,  to  be  accumulated  at  5  per  cent, 
per  ann.  compound  interest,  in  how  many  years  will  the 
debt  be  paid  ? 
ICOO 

ZI50.    Tlie  nearest  number  m  table  II.  preceding 

20 

page,  to  50  found,  is  51.1135,  and  the  number  answering 
to  this  is  26,  so  that  in  less  than  26  years  a  debt  of  1000/. 
would  be  extinguished  by  laying  by,  and  accunmlating,  at 
compound  interest,  annually  20/  per  annuiii.  If  the 
rate  of  interest  had  been  6  per  cent  24  years  would  have 
paid  the  debt,  but  at  4  per  cent,  it  would  have  taken  be- 
tween 28  and  29  years. 

Ex.  2.  How  long  will  75  guineas  a  year  be  in  accu- 
mulating to  2000/.  at  the  same  rate  ? 

Ans.  in  somewhat  less  than  17  years'. 

Ex.  3.  In  what  tim^e  will  an  annuity  of  25/.  amount 
to  3575/.,  at  the  same  rate  } 

^         Ans.  in  little  more  than  43  years. 

Ex.  4.  How  long  will  the  national  debt,  left  at  the 
time  of  Mr.  Pitt's  death,  viz.  581  millions,  be  in  paying 
off,  supposing  five  millions  annually  be  appropriated  for 
that  purpose,  and  the  rate  of  compound  interest  5  per 
cent.  ?  Ans.  in  less  than  40  year? , 

Ex.  5.  The  national  debt  was,  at  Midsummer  1807, 
756  millions  of  pounds,  out  of  which  the  commissioners 
had  redeemed  \\7  millions  and  a  half,  how  long  would 
the  remainder  take  in  paying  olf,  if  eight  millions  be  ap 
plied  annually,  at  the  rate  of  5  per  cent,  compound  inte- 
rest for  the  purpose  ?  Ans.  33  years, 


226  AND  ANNUITIES. 

II.    To  find  how  much  a  given  annuity  will  amount  to  in 
a  given  term,  at  5  per  cent,  compound  interest. 

Rule.  Multiply  the  given  annuity  by  the  number  in 
the  table  standing  opposite  to  the  given  term  of  years. 

Ex.  1.  I  can  lay. by  30/.  per  annum  with  its  interest; 
that  is,  i  can  appropriate  50/.  a  year  to  be  accumulated 
at  5  per  cent,  compound  interest,  how  much  shall  1  have 
saved  if  1  live  21  years  ? 

Opposite  to  21  years  I  find  S5.7192,  which  multiplied  by 
60,  gives  1785.9600.  Answer,  1783/.  19s.  2d. 

Ex.  2.  How  much  will  an  annuity  of  35/.  amount  to 
in  83  years  ?  Answer,  39461/.  9s.  lOjc/. 

Ex.  3»  To  what  sum  will  an  annuity  of  100  guineas 
amount  in  19  years,  at  5  per  cent,  compound  interest  ? 

Answer,  3206/.  12s. 

Ex.  4!  To   what  sum    will    60    dollars    per    annum 

amount  to  in  25  years,  at  5  per  cent,  compound  interest? 

Answer,  g2863,  62  cts.  6  mis. 

III.  The  PRESENT  VALUE  of  an  annuity  is  that  sum 
which,  if  improved  at  compound  interest,  would  be  suf- 
ficient to  pay  the  annuity. — For  this  the  following  table 
is  adapted. 


ANNUITIES. 


227 


TABLE  III. 

Shewing;  the  present  Value  of  an  Annuity  of  ll  for  any 
number  of  Years  not  exceeding  100,  at  5  per  cent,  per 
annum,  Compound  laterest. 


Yrs 

Value. 

Yrs. 

Value. 

Yrs. 

Valine.  jYrs.  | 

Value. 

1 

,952381 

26 

14,375183 

51 

18,338977 

76 

19,509495 

o 

1,859410 

27 

14,643034 

52 

18,418073 

77 

19,532853 

3 

2,723248 

:^S 

14,89-14/' 

53 

18,493403 

78 

19,555093 

4 

3,545950 

29 

1-),  14 1074 

54 

18,565146 

79 

19,576284 

5 

4,329477 

30 

15,372451 

55 

18,633472 

80 

19,596460 

6 

5,075692 

31 

15,592810 

56 

18,698543 

81 

19,615677 

7 

5,78637S 

32 

l.;,80. 677 

57 

18,7605  i  9 

82 

19,638978 

8 

6,453213 

33 

16,002549 

58 

18,819542 

83 

I9,65l4u7 

9 

7,107822 

34 

16,192901 

59 

18,875754 

84 

19,668007 

10 

7J21735 

15 

16,374194 

60 

18,9292&0 

85 

19,683816 

11 

8,306^14 

36 

16,546852 

61 

18,1^80276 

86 

19,698873 

12 

8,863252 

37 

.6.711287 

62 

19,028834 

87 

19,713212 

13 

9,393373 

38  16.86789;] 

63 

1j  075030 

83 

19,726869 

14 

9,898641 

39  17,01701-1 

64 

19,119124 

89 

19,739875 

15, 

10,379638 

40 

17,139086 

65 

19,161070 

90 

19,752262 

16 

10,8:37770 

41 

17,29456 

b6 

19,24019 

91 

19,764039 

17 

11,274066 

42 

17,42320'^ 

67 

19  239066 

92 

19,775294 

18 

11,689587 

4J 

17.54591: 

68 

19,275301 

93 

19,785994 

19 

12,085321 

44 

17,662773 

69 

19,309810 

94 

19,796185 

20 

12,462210 

45 

17,774070 

1  70 

19,342677 

95 

19,805891 

21 

12,821153 

46 

17,880066 

71 

19,37397S 

96 

19,815134 

22  1 13, 16  "^003 

47 

17,981016 

72 

19,403788 

97 

19,823937 

23 

13,488574 

48 

18,077158 

73 

19,4321.-9 

98 

19,832321 

24 

13,798643 

49 

18,168722 

74 

19,459318 

99 

19,840306 

25 

4(.>   93945|  50 

18,253925 

75 

19,484-70 

100 

19,847910 

To  find  the  present  value  of  an  annuity  for  a  term  of  years. 

Rule.  Multiply  the  number  in  the  table  opposite  to 
the  given  term  of  years,  by  the  sum,  and  the  product  is 
the  answer. 

Ex.  I.  What  is  the  present  value  of  an  annuity  of 
126^.  for  21  years  ? 

In  the  table  opposite  21  is  12.8!2ll53;  this  multiplied 
by  126,  gives  1615.465278  ZZ  1615/.  9s.  Si. 


228  OHAXCES. 

Ex.  2.  WItat  is  the  present  value  of  an  annuity  of  75 
dollars  for  12  3 ears,  at  5  percent.  ? 

Answer,  8664:74.39. 

Ex.  3.  What  present  sum  is  equivalent  to  a  nett  rent 
©f  45/.  per  annum  for  84  years,  allowing  interest  of 
money  at  5  per  cent.  ?  Answer,  883^  nearly. 


CHANCES.* 


Q_uestlon  /.—Suppose  a  counter,  having  a  black  and  a 
white  face,  be  thrown  up,  to  see  which  will  be  upper- 
most, after  the  counter  has  fallen  to  the  ground^  and  if 
the  whit^'  face  appear  uppermost,  a  person  is  to  have  5 
shillings,  what  is  the  chance,  or  probability,  that  he  will 
be  entitled  to  the  five  shillings  ? 

Solution.  Since  either  the  black  or  the  white  face 
must  be  uppermost,  there  is  an  equal  chance  for  the  ap- 
pearance of  either  face,  of  course  the  chance,  or  the  pro- 
bability, may  be  expressed  by  ~,  or  a  bystander  ought  to 
give  him  2s.  6d.  for  his  chance  of  getting  the  five  shil- 
lings. 

(Question  II. — Suppose  there  are  three  counters  put 
into  a  bag,  one  red,  another  white,  and  a  third  black; 
out  of  whicli.if  aperson  blindfolded  take  the  red  he  is  to 
have  5  shillings,  1  demand  the  value  of  t.ie  chance,  or 
what  is  the  probability  of  his  drawing  the  red  counter  } 

*  It  is  meant  only  to  give  so  much  of  the  doctrine  of  chances, 
as  shall  enable  the  pupil  to  understand  upon  vvliat  ground  the 
doctrine  of  Anmiities,  8ic.  depends.  To  illustrate  this  part  of 
the  subject,  recourse  will  be  had  to  some  familiar  instances, 
vjrhich  may  seem,  at  first  sight,  to  lead  to  gaming  ;  but  it  is  be- 
lieved, that  the  facts  adduced  must,  if  properly  considered,  de- 
ter young  persons  from  tins  pernicious  and  destructive  vice, 
which  IS  too  much  encouraged  by  the  almost  perpetual  drawing 
of  state  lotteries. 


CHANCES. 


229 


'Solution.  lie  has  evidently  one  chance  out  of  three, 
and  therefore  the  probability  may  be  valued  at  •},  and 
another  person  inclining  to  purchase  his  chance,  ought 
to  give  for  it  the  ^d  of  5  shillings,  or  Is.  Sd. 

In  the  former  case,  the  chances  for  the  event's  hap- 
pening and  failing  are  equal,  and  each  being  equal  to  |, 
the  certainty  is  reckoned  as  1,  or  unity. 

In  this  last  ease,  there  is  one  chance  for  the  event's 
happening,  and  two  for  its  failing  :  in  other  words  the 
chance  for  its  happening  is  -J,  and  for  its  failing  -|:  here, 
again,  the  chances  for  the  happening  and  failing  are  equal 
to  unity,  because  ^  +  f  iz  |  z:  1. 

Question  111. — Suppose  there  are  five  counters,  two 
white  and  three  black,  out  of  which,  when  mixed,  a  per- 
son blindfolded  is  to  draw  one  of  the  white,  and  in  that 
case  is  to  be  entitled  to  5s.,  what  is  his  chance  for  so 
doing,  and  what  is  his  expectation  worth  ? 

Solution.  It  is  plain  here  are  five  chances  in  the 
whole,  of  which  there  are  two  only  out  of  five  for 
taking  a  white  counter,  and  the  other  three  for  taking 
a  black  one  ;  therefore  the  probability  of  winning  may 
be  expressed  by  the  fraction  f,  and  of  missing  -^,  ai:d 
he  might  sell  his  expectation  of  the  five  shillings  for 
fths  of  that  bum,  that  is,  for  two  shillings. 

Ex.  J.  At  the  conclusion  of  the  last  state  lottery, 
when  there  were  only  five  tickets  left  in  the  wheel, 
there  were  two  prizes  oli  50^  each,  and  three  blanks, 
what  was  the  value  of  one  of  those  tickets  ? 

'  Answer,  £0/. 

Ex.  2.  What  is  the  value  of  one  ticket  when  only 
five  are  left  in  one  wheel,  and  in  the  other  there  is 
one  prize  of  100/.  and  four  blanks  .^^ 

Answer,  201. 

Ex.  S.  AVhat  chance  has  the  holder  of  a  single  lottery 
ticket  of  a  prize,  when  there  are  three  blanks  to  a 
prize  ?  Answer,  4  to  1. 

20 


230  CHANCES. 

.  Question  IV. — What  is  the  prohability  of  throwing 
an  ace  with  a  single  die,  in  one  trial  ? 

Solution.  There  are  six  faces  to  a  die,  of  which  one 
only  is  the  ace,  therefore  the  probability  of  throwing  an 
ace  with  a  single  die  in  one  trial  is  expressed  by  \  ;  and 
the  probability  of  not  throwing  an  ace  is  |  :  here,  as 
before,  the  chances  for  not  throwing  the  ace,  and  that 
for  throwing,  are  together  equal  to  unity. 

Question  V. — AVhat  is  the  probability  of  throwing  an 
ace  in  four  throws  ? 

Solution.  "We  must  consider  the  probability  of  fail- 
ing in  the  four  tlirovvs.  'J  he  probability  of  missing  the 
first  lime  will  be  -^  ;  so  it  is  the  second,  third,  and  fourth 
times  ;  therefore  the  probability  of  missi.ig  in  all   four 

5  5         5         5  625 

throws  will  be— X— X  —  X— .  = ;  which  sub- 

6  6         6         6         J  296 

12^6—625         67  \ 

tracted  from  unity  or  J,  gives = ,  which 

1296  1^96 

is  the  probability  of  throwing  it  once  or  oftener  in  four 
turns  ;  therefore  the  odds  of  throwing  an  ace  in  four 
times,  is  as  67  J  to  025,  or  rather  more  than  an  even 
chance* 

The  probability  in  three  throws  will  be, 

5         5        5  125         216—125  91  - 

6^6         d  £1()  216  216 

Here  the  odds  is  against  throwing  the  ace  in  three 
throws,  as  91  is  less  tlian  125. 

question  VI.  In  two  heaps  of  cards,  one  containing 
the  13  diamonds,  the  other  the  13  spades,  placed  promis- 
cuously, what  is  the  probability  that,  takiiig  one  card  at 
a  venture,  out  of  each  heap,  1  shall  take  out  the  two 
aces  ? 

Solution.  The  probability  of  taking  the  ace  out  of  the 
first  heap  is  ^V  ?  ^^®  probability  of  taking  the  ace  out  of 


E5i:PECTATI0N    OF    LIFE.  231 

le  second  heap  is  also  -Jj,  therefore  the  probability  of 

iking  out  both  aces  is  -j\  x  -jV'  ^^ »  vvhich  sub- 

169 
168 

tracted  from  I,  gives ,  of  course  the  chances  against 

109 
me  are  as  168  to  1  :  in   ojier  words,  I  may  eji'pect  to  do 
this  once  in  169  attempts. 

On  similar  principles  the  evpectntinn  of  life  is  found. 
It  is  known  by  accurate  observation,  tliat  of  46  per- 
sons aged  40  years,  one  will  die  every  year,  till  they 
are  all  dead  in  46  years  ;  therefore  half  46,  or  23  years, 
will  be  the  expectation  of  life  of  a  person  40  years  of 
ao;e.  That  is,  the  number  of  years  enjoyed  by  them 
all,  will  be  just  the  same  as  if  every  one  of  them  had 
lived  23  years,  and  then  died.  The  same  reasoning  ap- 
plies to  all  other  aijes,  which  leads  us  to  a  more  particu- 
lar consideration  of  the  subject. 


EXPECTATION  OF  LIFE. 


From  the  Bills  of  Mortality  in  different  places,  tables 
have  been  constructed  which  shew  how  many  persons, 
upon  an  average,  out  of  a  certain  number  born,  are  left 
at  the  end  of  each  year  to  the  extremity  of  life.  From 
such  tables,  wliich,  as  we  have  seen,  are  founded  on  the 
doctrine  of  Chances,  the  probability  of  tiiexontinuance 
of  a  life,  of  any  proposed  a^e  is  known. 


232 


EXPECTATION    OF   LUTE. 


TABLE  I. 

Shewing  the  Probabilities  of  tbe  Duration  of  Hmnaii 
Life,  deduced  from  the  Register  of  Mortality  at 
Northampton. 


Persons 

Dec. 

iVrsoiiB 

IXc. 

PevsoJis 

1  Dec 

Age 

living-.  ' 

of  Life 

Age 

living. 

of  Life. 

75     1 

Age. 

06 

living 

of  Life. 

0 

il6iu 

4J-'0 

1552 

80 

1 

8650 

13(S7  ' 

34 

4083 

75 

67 

1472 

80 

2 

723; 

502 

.15 

.'AO 

75 

68 

1392 

80 

3 

67  SI 

335 

36 

3935 

75 

69 

1312 

89 

4 

6U6 

197 

37 

3860 

75 

70 

1232 

80 

5 

6249 

184 

38 

3785 

75 

71 

1152 

80 

6 

&'65 
1925 

140 

..9 

3710 

75 

72 

1072 

80 

7 

110 

40 

3685 

76 

73 

992 

80 

8 

58:5 

80 

41 

3559 

77 

74 

912 

80 

9 

r>7r>5 

60 

42 

3^82 

73 

75 

832 

80 

10 

5G75 

52 

43 

3404 

7S 

76 

752 

77 

11 

562.) 

50 

44 

3326 

78 

77 

675 

73 

12 

5f,7S 

5U 

45 

3248 

78 

78 

602 

68 

13 

5523 

50 

46 

3170 

78 

79 

534 

65 

14 

5473 

50 

47 

3092 

78 

^^o 

469 

63 

15 

5423 

50 

48 

3014 

78 

81 

406 

60 

16 

5373 

5:i 

49 

2936 

79 

8.> 

346 

57 

17 

5320 

58 

50 

n57 

81 

83 

289 

55 

18 

5262 

C3 

51 

2776 

82 

84 

234 

48 

19 

5199 

67 

52 

2694 

82 

S5 

186 

41 

20 

5132 

72 

53 

2612 

82 

86 

145 

34 

21 

5.6'J 

75 

54 

2530 

82 

87 

lit 

28 

22 

49 -,55 

75 

-55 

2448 

82 

88 

83 

21 

2li 

4910 

73   ' 

56 

2366 

82 

89 

62 

M> 

24 

4835 

75 

57 

2281 

82 

9,) 

40 

12 

25 

-  4760 

75 

58 

2202 

82 

91 

3i 

10 

26 

4685 

75 

5? 

2120 

82 

92 

24 

8 

27 

4610 

75 

e^j 

2038 

82 

93 

16 

7 

23 

45J5 

75 

61 

1956 

82 

94 

9 

5 

29 

4460 

75 

62 

1874 

81 

95 

4 

3 

30 

43S5 

75 

63 

1793 

81 

96 

1 

1 

Si 

4310 

75 

64 

1712 

80 

3i 

4235 

75 

65 

1632 

80 

Case  L    To  find,  by  this  Table,  the  expectation  of  any 
single  life. 

.Rule.     Divide   tbe  sum  of  all  the  living  in  the  table, 
at  the  age  whose  ex[>ectution  is  required,  and  at  all  great* 


EXPECTATION    OF    LIFE.  23S 

cr  ages,  bv  the  sum  of  all  that  die  annually  at  that  age, 
and  ab(>ve  it,  or.  wiiicli  is  the  same  thinj^,  by  the  number 
in  tlje  table  of  the  living  at  that  a^^e,  and  half  unity,  or  .5 
subtracted  from  the  quotient  will  be  the  expectation  re- 
quired. 

Ex.  1.  What  is  the  expectation  of  a  life  at  60  ? 

The  sum  of  the  living  at  the  age  of  60  and  upwards, 
by  the  table,  is  27947,  which  divided  by  2038,  the  num- 
ber of  lining  at  that  age,  gives  13.71,  from  which  subtract 
.5,  and  the  expectation  of  a  life  at  60  is  equal  to  13.2 J,  or 
13  years  1 1  weeks  nearly. 

Ex.  2.  What  is  the  expectatin  of  a  life  70  years  of 
age,  one  of 80,  and  one  of  90  ? 

Ans.  life  of  70  is  8  years  31  weeks — Hfe  of  80  is  4|. 
years — life  of  90  is  2  years^  20  weeks,  and  5  days. 

Case  II.  To  find  the  probability  that  a  given  life  shall 
continue  any  number  of  years,  or  attain  a  given  age. 

Rule.  Make  the  number  in  the  table,  opposite  to  the 
proposed  age,  the  numerator  of  the  fraction,  and  for  the 
denominator  take  the  number  opposite  the  present  age. 

Ex.  I.  What  is  the  probability  that  I,  who  am  45,  shall 
live  to  60  .^ 

The  number  against  60ii:2038  "|  Therefore  the  chances  in 

Lmy  favour  are  20   :    12 

The  number  against  45IZ3248  J  nearly,  or  as  -  5  :  3. 

2038 

For,  since  the  probability  of  living  is  equal  to ,  the 

3248 
chance  of  dying  during  that  period  is 
2038     3248—2038      1210 

1 -zz — iz .    The  denominators  being 

3248  3248  3248 

the   aiiie,  tho  chance  of  life  is  to  the  probability  of  dying 
as  2038  to  1210,  or  as  20  to  12,  or  as  5  to  3  nearly. 

go* 


23  i 


EXPEcrATION    OF    LIFE. 


Ex.  2.  What  is  the  probability  that  a  peiison  aged  21, 
as  lattain  to  54  ?  2530 

Ans. chance  of  living: 

5060 
Ex.  3.  What  is  the  probability  that  a  person  aged  15 
should  live  till  70  ?  1232 

Ans. chance  of  living. 

5423 
Ex.  4.  What  chance  has  a  person  aged  70  of  living  10 
years  longer  ?  469 

Ans.    — ^  chance  of  living. 
1232 
From  the  foregoing  table  is  formed 

TABLE  H. 

Shewing  the  expectation  of  Human  Life  at  every   Age 
according  to  the  Probabilities  found  bv  Table  I. 


Age. 

Expectation. 

Age. 

Expectation, 

Age. 
50 

Epoctation. 

Ag.-. 

Expectatioi». 

0 

25,18 

25 

o0.85 

17,99 

75 

6,54 

1 

32,74 

26 

30,33 

51 

17,50 

76 

6,18 

2 

37.79 

27 

29,82 

52 

17,02 

77 

n,S3 

3 

39,55 

28 

29.3:; 

53 

16,54 

78 

5,48 

4 

40,58 

29 

28,79 

54 

16,06 

79 

5,11 

5 

40,84 

30 

28  27 

53 

15,58 

80 

4,75 

6 

43,07 

31 

27,76 

55 

15,10 

81 

4,41 

7 

41,08 

32 

27,24 

57 

14,63 

82 

4,09 

8 

40,79 

33 

2e,72 

58 

14,15 

83 

3,80 

9 

40,36 

34 

26,20 

b9 

13,68 

H4 

3,58 

10 

39,78 

35 

25.6^ 

60 

l3,2l 

85 

3,37 

11 

39,14 

36 

25,16 

6] 

12,75 

86 

3,19 

1$ 

3a. 49 

37 

24,64 

62 

12,28 

87 

3,01 

13 

37,83 

38 

24,  12 

63 

11,81 

88 

'2,S6 

14 

37,17 

39 

23,.0 

64 

11,35 

89 

2,66 

15 

36,51 

40 

23,03 

65 

10,88 

90 

2,41 

16 

35,85 

41 

22,56 

66 

10,42 

9". 

2,09 

17 

35,20 

42 

22,!  4 

67 

9,96 

92 

1,75 

18 

34  58 

43 

21,54 

68 

9,50 

93 

1,37 

19 

33,99 

4i 

21,03 

69 

9,05 

94 

1,05 

20 

33,43 

45 

20,52 

70 

8,60 

95 

0,75 

31 

32,90 

IG 

20.02 

71 

8,17 

96 

0,50 

22 

32,39 

47 

19,51 

72 

7,74 

23* 

31,88 

48 

19,00 

73 

7,^3 

^k 

31,36 

49 

18,49 

r4 

6,92 

LIFE    ANNUITIES.  233 

To  find  the  expectation  of  any  given  life. 

Rule.  Seek  in  the  table  the  given  age,  and  oppo- 
site to  it  is  the  expectation. 

Thus,  the  chance  of  life  to  an  infant  just  horn  is  25.18, 
or  rather  more  than  25  years  ;  to  a  person  of  45  years  of 
aire  20,52,  as  we  have  found  before,  and  to  a  person 
of  69,  just  9  years. 


Upon  these  tables  is  found^^d  the  doctrine  of 

LIFE  ANNUITIES. 

Life  Aknuities  are  annual  payments  to  continue 
dufins:  any  life  or  lives.  The^^e  are  generally  purchased 
or  s>!ld  for  a  present  sum  of  money. 

**  The  present  value  of  a  life  annuity**  is  the  sum  that 
would  be  sufficient,  (allowins:  for  the  chance  of  life  failing, 
which  has  been  considered  in  the  preceding  pages)  to 
pay  the  annuity  without  loss. 

If  money  bore  no  interest,  the  value  of  an  annuity  of 
1/  would  be  equal  to  the  expectation  of  life-  Thus, 
Table  [I.  the  value,  of  an  annuity  for  a  life  of  20 
years  of  age,  if  money  bore  no  interest  would  be  equal  to 
nearly  S3  years  and  a  half  purchase  ;  that  is  3S/.  los. 
in  hand  for  each  life,  would  be  sufficient  to  pay  to  any 
number  of  suc!^  lives  iL  per  annum. 

If  money  is  capable  of  being  improved  bv  being  put 
out  to  interest,  tlie  sum  just  mentioned  would  be  more 
th.in  the  value,  because  it  would  be  more  than  sufficient 
to  pay  the  annuity  ;  and  it  will  be  as  much  more  than 
sufficient  as  the  interest  is  greater.     As  an  example. 

If  money  can  be  improved  at  5  per" cent,  compound 
iviterest,  the  half  of  33',  10s.,  or  16/  15s.,  will,  as  we 
have  seen,  in  little  more  than  14  years,  produce  the  33^ 
lO-.  required. 

It  must  not  however  he  supposed,  that  16^  I5s  is  the 
true  value  of  an  annuity  of  il.  during  a  life  of  20.    The 


236 


LIFE    ANNUITIES. 


value  of  an  annuity  certain  for  a  term  equal  to  the  ex- 
pectation, always  exceeds  the  true  value,  because,  In  a 
number  of  life  annuities,  many  of  the  payments  would 
not  be  to  be  made  till  a  much  more  remote  period  than 
the  term  equal  to  the  expectation. 

Upon  this  principle  the  following  table  is  computed, 
from  which  it  appears  that  the  present  value  of  an  annui- 
ty of  1/.  on  a  life  of  20  years  of  age,  is  equal  to  14/.  and 
a  small  fraction  only  ;  that  is,  14/.  in  hand  for  each  life, 
improved  at  compound  interest,  will  be  sufficient  to  pay 
to  any  number  of  such  lives  1/.  per  annum. 

TABLE  I. 

Shewing  the  Value  of  an  Annuity  of  U,  on  a  Single  Life, 
at  every  Age,  according  to  the  probabilities  of  the 
Duration  of  Human  Life  at  Northampton,  reckoning 
interest  at  5  per  cent. 


Age. 

Value. 

Age 

25 

1     Value. 

Age. 
'   50 

Value. 

Age. 
75 

Value. 

Birth. 

8  863 

13.567 

10,269 

4,744 

1  year 

11.563 

26 

13473 

1    ^1 

1C,097 

76 

4,511 

^2 

13.420 

2? 

13.377 

52 

9,925 

77 

4,277 

3 

14.135 

2« 

13.278 

53 

9,748 

78 

4,035 

4 

1k613 

29 

1.3.177 

54 

9,567 

79 

5,776 

5 

14  827 

30 

13.072 

55 

9,382 

80 

3,515 

6 

15.041 

31 

12.965 

56 

9,193 

81 

3,263 

7 

15166 

32 

12.354 

57 

8,999 

82 

3,020 

8 

15.226 

33 

12  740 

58 

8,801 

83 

2,797 

9 

15.210 

34 

12.623 

59 

8,599 

84 

2,627 

10 

15139 

35 

12.502 

60 

8,392 

85 

2,471 

11 

15.043 

36 

12.377 

61 

8,181 

86 

2,328 

.12 

14.937 

37 

12.249 

62 

7,966 

87 

2,193 

13 

14.826 

38 

12.116 

63 

7,742 

88 

2,080 

14 

14.710 

39 

11.979 

64 

7,514 

89 

1,924 

15 

14.588 

40 

11,837 

'  65 

7,276 

90 

1,7^3 

16 

14.460 

41 

11,695 

'   6b 

7,034 

91 

1,447 

17 

14.334    1 

42 

11,551 

67 

6.787 

92 

1,153 

18 

li.2J7    • 

43 

11,407 

68 

6,536 

93 

0,ii\6 

19 

14.108 

44 

11,258 

69 

6,281 

94 

0,524 

20 

14.007 

45 

11,105 

70 

6,023 

95 

0,238 

21 

13917 

46 

10,94? 

n 

5,764 

96 

0,000 

22 

13.83  i 

47 

10,784 

72 

5,504 

23 

13.746 

48 

10,616 

73 

6,245 

24    1 

13.658 

4y 

10,44  -. 

74 

4,990 

4-IFE    ANNUITIES.  237 

I 

To  find  the  value  of  an  annuity  for  a  person  of  any- 
given  age. 

Rule.  Multiply  the  number  in  the  table  against  the 
given  age,  by  tlie  sum,  and  the  product  is  the  answer. 

Ex.  1.  What  shouhl  a  person,  aged  45,  give  to  pur- 
chase an  annuity  of  60^  per  antium  during  life,  interest 
being  reckoned  3  per  cent  ? 

The  value  in  the  table  against  45  years  is  U.  105,  and 
this  multiplied  by  00  gives  the  answer,  066/.  6s. 

Ex.  2.  A  person  aged  69  years  would  purchase  an 
annuity  of  200/.  for  life,  what  must  he  pay  for  it  in  ready 
money  at  the  same  rate  of  interest  ? 

Answer,  1256^  4s. 

Ex.  3.  A  merchant  marries  a  lady  aged  28,  whose  for- 
tune for  life  is  300/.  per  annum,  being  desirous  of  con- 
verting the  same  into  money,  what  ought  he  to  have  for 
itj  allowing  interest  5  per  cent.  ?         Answer,  3983/.  8s. 

Ex.  4.  What  is  the  value  of  an  annuity  of  200  dollars 
during  the  life  of  a  person  aged  25  years  ? 

Answer,  552713  40  cts. 

Ex.  5.  What  is  the  value  of  50/.  per  annum,  payable 
during  the  life  of  a  person  aged  41  years  ? 

Answer,  584/.  15s. 

Ex.  6.  What  is  the  value  of  a  clear  annuity  of  75/. 
during  the  life  of  an  old  man  aged  76  ? 

Answer,  338/.  6s.  Qd. 

Ex.  7.  What  is  the  value  of  a  landed  estate  during 
the  life  of  a  person  aged  38,  produing  nett  3o/.  9s.  per 
annum?  Answer, 3(J8/.  IBs. 7^fi/. 

Ex.  8.  What  is  the  life  interest  of  a  person  aged  53, 
in  1250/.  3  per  cent.  Consols  worth  .^ 

Answer,  3G5/.  lis. 

Ex.  9.  A  gentleman  aged  60,  who  receives  an  annuity 
of  150/.  per  annum,  for   life,  out  of  a  freehold  estate^ 


238  LIFE    ANNUITIES. 

wishes  to  exchangie  his  life  for  that  of  his  wife,  aged  32  ; 
what  ought  to  be  required  of  him  for  so  doing  ? 

Answer,  669^,  6s. 

Ex.  lO.  A  person  having  an  annuity  of  lOOl.  during  a 
life  of  37  years,  agrees  to  exchange  it  for  an  equivalent 
annuity  during  a  life  of  45  ;  what  annuity  should  be 
granted  him  ?  Answer,  ilOZ.  6s. 

Ex.  11.  What  annuity  will  \00l.  purchase  during  the 
life  of  a  person  aged  28  .^  Answer,  7^..  10s.  7c?. 

Ex.  12.  A  parish  means  to  raise  a  sum  of  money  for 
building  a  workhouse,  by  life  annuities ;  at  what  ages 
should  they  grant  7,  8  and  9  per  cent.  ?* 

Ans.      To  persons  of  18,  35  and  45  years  of  age. 

Ex.  13.  What  is  the  difference  in  value  between  an 
annuity  of  40/.  during  a  lifeof36,  and  an  annuity  certain 
for  20  years  Pf 

Answer,  3l.  8s.  Qd.  nearly. 

Ex.  14.  What  annuity  should  be  granted  to  a  person 
aged  57  during  his  life,  for  2,000Z.  five  pes  cent,  stock, 
which  is  now  at  99|  ?  Answer,  22U.  8s. 

NOTES. 

*  Questions  of  this  sort  are  answered  by  dividing  \00l. 
by  the  rates  per  cent.,  and  opposite  to  the  numbers  in 
the  table  that  are  nearest  the  quotient,  are  the  required 
ages  :  thus,  to  find  at  what  age  a  life  annuity  of  9  per 

KG 
cent,  should   be  granted, —  =  11.111  :  the  nearest 

9 
number  in  the  table  is  1 1.105,  by  the  side  of  which  is  45, 
hence,  to  ages  of  45,  an   annuity  of  9  per  cent,  may  be 
granted. 


LIFE    ANNUITIES. 


239 


TABLE  II. 

Shewing  the  Value  of  an  Annuity  during  the  joint  con- 
tinuance of  Two  Lives,  accorciing  to  the  probabilities 
of  Life  at  Northampton,  reckoning  interest  at  5  per 
cent. 


Ages, 

Value. 

Ages. 

Value. 

Ages. 

Value. 

Ages. 

Value. 

5-5 

11,984 

15-35 

10,655 

30^0 

10,255 

45-70 

5,195 

5-10 

;  2,315 

15-40 

10,205 

30-35 

9,954 

45-75 

4,206 

5-15 

11,954 

15-45 

9,690 

30-4U 

9,576 

45-80 

.:>,197 

5-20 

ri,5f)l 

15-50 

9;076 

30-45 

9,135 

50-50 

7,522 

5-25 

11,281  . 

15-55 

8,403  ' 

30-50 

8,596 

50-55 

7,098 

5-30 

10,959 

15-60 

7,022 

30-55 

7,999 

50-60 

6,568 

5-35 

10,5/2 

i  5-65 

6,705 

30-60 

7,292 

50-65 

5,89r 

5-40 

10,H)2 

15-70 

5,r)31 

30-65 

6,447 

60-70 

5,054 

5-45 

9,571 

15-75 

4,495 

30-70 

5,4^^2 

50-75 

4,112 

5-50 

«,94l 

15-80 

3,372 

3U-75 

4,365 

50-80 

3,140 

5-55 

8,256 

20-20 

11,232 

30-80 

3,290 

,55-55 

6,735 

5-60 

7  406 

20- .^5 

10,989   ( 

35  35 

9.680 

55-60 

6,272 

5-u5 

6.546 

20-30 

10,707  ' 

35-40 

9,331 

55-65 

5,671 

5-70 

5,472 

20-35 

10,363   : 

35-45 

8.l"zl 

55-70 

4,893 

5-75 

4,362 

20-40 

9,9j7   ' 

35-50 

8415 

55-75 

4,006 

5-80 

3,238 

20-45 

9,4+8 

35-55 

7,849 

55-80 

3,076 

10-10 

12,665 

20-5'J 

8,801 

35-60 

7,174 

60-60 

5,888 

10-15 

12,302 

20-55 

S,216 

35  65 

6,360 

60  65 

5,372 

10-20 

11,906 

20-60 

7,463 

35-70 

5,382 

60-70 

4,680 

10-25 

11,627 

20-65 

6,576 

35-75 

4,3^7 

60-75 

3,866 

l'y-50 

11,304 

20-70 

5,532 

35-80 

3,268 

60-80 

2,992 

10-35 

10,916 

20-75 

4,4.-4 

40-40 

9,016 

65-65 

4,960 

10-40 

10,442 

20-80 

3,325 

40  45 

8,643 

65-70 

4,378 

10-45 

9,900 

25-25 

10;764 

40-50 

8,171 

65-75 

3,iJ65 

10-50  1  9,2C0 

26-30 

10,499 

40-55 

7,654 

65-80 

2,873 

10-55 

8.5(30 

25-35 

10,175 

40  60 

7,015 

70-70 

3,930 

lo-fiu 

7,750 

25-40 

9,771 

40-65 

6,240 

70-75 

3,347 

10-05 

6,803 

25-45 

9,301 

40-70 

5,298 

70-80 

2,675 

10-70 

5,700 

25-50 

8,73^ 

40-75 

4,272 

75-75 

2,917 

10-75 

4,522 

25-55 

8,116 

40-80 

3,236 

75-80 

2,381 

10-80 

3,395 

25-60 

7,383 

45-45 

8,512 

80-80 

2  01!| 

15-15 

11,960 

25  65 

6,515 

45-50 

7,891 

85-85 

1,256 

1 5-20 

11,^85 

25-70 

5,489 

45-55 

7.  ill 

90-90 

0,909 

15-25 

11,324 

1  25^75 

4,390 

45-60 

6,822 

15-30  111,020 

25-80 

3,308 

1  45-65 

6,094 

Case  I.   To  find  the  value  of  an  annuity  on  the  longest 
of  two  single  lives. 

Rule.  From  the  sum  of  the  values  of  the  single  lives, 
subtract  the  value  of  their  joint  continuance,  and  the  re- 
mainder will  give  the  value  of  the  longest  of  the  lives. 


240  LIFE  annuities: 

Ex.   1    What  is  the  value  of  the  longest  of  two  lives 
aged  10  and  15  ? 

ri'  11    T   S  The  value  of  a  life  at     -      -      10=15.159 
lablel.^ .    .      .      15  =  14.588 


29.727 
♦   Table  II,  The  value  of  the  joint  continuance 

of  two  lives  of    -     -     10  and  15  =  12302 


Value  of  Ihe  longest  of  the  two  lives   17.425 
Therefore  an  annuity  ot  iCO/.  a  year  upon  the  longest  of 
two  lives,  one  10  and  the  other  15,  would  be  worth  near- 
ly 17  years  and  a  iialf  purchase,  or  more  accurately, 
1742^.  10s.' 

Ex.  2.  What  is  the  value  of  an  annuity  on  ihi  longest 
of  two  lives  whose  ages  are  thirty  and  forty  ? 

Answer,  1533/.  6s. 

Case  II.  To  find  the  value  of  an  annuity  on  three  joint 
lives. 

Rule.  Take  the  value  of  the  two  elder,  and  find  the 
age  of  a  single  life  equal  to  that ;  then  find  the  value  of 
the  joint  lives  of  this  now  found)  and  the  youngest 

Ex.  I.  Let  the  three  lives  be  20,  30,  and  40. 

The  value  of  the  joint  continuance  of  the  two  eldest ; 
viz.  of  SO  and  40  (by  Table  II.)  is  equal  to  9.576,  which 
answers  to  a  single  life  (by  Table  1.)  of  54.  Now,  the 
value  of  the  joint  lives  of  20  and  54  by  Table  IF.,  or  the 
ages  which  come  nearest,  viz.  20  and  55,  is  8  210*  for 
the  value  sought :  hence  an  annuity  of  40/.  on  three 
joint  lives  would  be  worth  about  328/.  12s. 

Ex.  2.  To  find  the  value  of  3  joint  lives  of  the  ages 
15,  30,  and  45.  Answer,  8.403. 

NOTE. 

*  The  numbers  9.576  and  8.216,  are  not  quite  ac- 
curate, because  the  limits  of  this  book  do  not  admit  of  a 
table  giving  the  combinations  of  all  ages. 


LIFE    ANNUITIES.  241 

Ex.  3-  \Miatis  the  value  of  an  annuity  of  150/.  on  the 
joint  continuance  of  three  lives  of  the  ages  50,  60,  and 
70  ?  Answer,  587/.  14s. 

Case  III.  To  find  the  value  of  the  longest  of  any  three 
lives. 

Rule.  From  the  sum  of  the  values  of  all  the  single 
lives  subtract  the  sum  of  the  values  of  all  the  joint  lives, 
combined  two  and  two.  To  the  remainder  add  the  value 
of  the  three  joint  lives,  and  the  sum  will  be  the  value  of 
the  longest  of  the  three  lives. 

Ex.  1.  What  is  the  value  of  the  longest  of  three  lives, 
whose  Bges  are  20,  30,  and  40  ? 

rvalue  of  a  life  of  20  =  14.007 

Table  1.4      _    —     —       30  =  13.072 

(^    '—     —    —       40=  11.837' 

38.916 

Value  of  two  joint  lives  of  20  and  30  =  i0.ro7 

—    —    —  20  and  40  =    9  937 

—    —     —    —    —     —  30  and  40  =  ^9.376 

38.916 ■ 

S0.22O  30.220 


8.096-f8.216  (the  value  of  the  joint  lives  found  in 
Ex.  1,  Case  II.)  =  16.912  =  the  value  of  the  longest  of 
the  three  lives. 

Ex.  2.  What  is  the  value  of  the  longest  of  three  lives, 
■whose  ages  are  15,  30,  and  45  ?  Ans.  17.126. 

Ex.  3.  What  is  the  value  of  an  annuity  on  the  longest 
of  three  lives,  whose  ages  are  50,  60,  and  70  ? 

Answer,  12.300. 

EXAMPLES   FOR    PRACTICE. 

Ex.  1.  What  is  the  present  value  of  an  annuity  of  50^, 
on  the  joint  lives  of  two  persons,  each  30  years  of  age  > 

Answer,  61 2^»  Ids'. 


242  LIFE    ANNUITIES. 

Ex,  2.  What  is  the  present  value  of  an  annuity  of  65^., 
during  the  joint  lives  and  the  life  of  the  survivor,  of  a 
man  aged  45,  and  his  wife  aged  35  ? 

Answer,  954/.  12s.  nearly. 

Ex.  3.  What  is  the  value  of  a  lease  producing  27Z. 
13s.  per  annum,  on  the  longest  of  two  lives  aged  60  and 
45  ?  An3wer,  3501.  9s, 

Ex*  4.  What  is  the  value  of  an  annuity  of  4-0/,  on  two 
joint  lives  of  70  and  5  years  ?  Ans.  218/.  I7s.  7d, 

Ex.  5.  What  is  the  value  of  an  annuity  of  50/.  on  the 
longest  of  two  lives  of  70  and  5  years  ? 

Answer,  768/.  18s. 

Case  IV.  To  find  the  value  of  an  annuity  on  a  given  life 
for  any  number  of  years. 

Rule.  Find  the  value  of  a  life  as  many  years  older 
than  the  given  life  as  are  equal  to  the  term  for  which  the 
annuity  is  proposed.  Multiply  this  value  by  ll.  payable 
at  the  end  of  this  term,  and  also  by  the  probability  that 
the  life  will  continue  so  long.  Subtract  the  product  from 
the  present  value  of  the  given  life,  and  the  remainder 
multiplied  by  the  annuity  will  be  the  answer. 

Ex.  1.  What  IS- the  value  of  an  annuity  of  50/.   per 
annum,  for  14  years,  on  a  life  of  35  ?    35  +  14  =  49. 
The  value  of  a  life  of  49  (14years  older  than 

the  given  life,  by  Table  1.)    -    -     -    -     =      10.443 
The  value  of  1/.   payable  at  the  end  of  14 

vears  (Table  ) =  .505068 

TH^e  probability  that  a  life  of  35  will  con- 1  2936 

tinue    14  years  (Fable  and  the  2d  >.  ^ 

Case.) J  4010 

10.443  X  .505068  X  (?^^  .7322  =3.861,  which,  sub-. 
\40i0/  ' 

tracted  from  12.502,   the  value  of  a  life  of  35,  Table  I. 

gives  8.641 ;  and  8.641  X  50  =  432/.  Is. 

Ex.  2.  What  is  the  value  of  an  annuity  of  80/.  per 
annum  for  20  years  provided  ^a  person  aged  45  live  so 
long  ?  '    Answer,  785/.  13s.  7d, 


LIFE    ANNUITIES. 

TABLE. 


Shewing  the  present  Value  of  H.  to  be  received  at  the 
end  of  any  number  of  years,  not  exceeding  100  ;  dis- 
counting at  5  per  Cent.  Compound  Interest. 


Yrs. 

Value 

Yrs. 

26 

Value. 

Yrs. 

Value. 

Yrs. 

Value. 

1 

.95i38J 

.281241 

5i 

.083051 

76 

.024525 

2 

.907029 

27 

.267848 

52 

.079096 

77 

.023357 

3 

.863838 

28 

.255094 

53 

.075330 

78 

.022245 

4 

.822702 

29 

.242946 

54 

.075743 

79 

.021186 

5 

.783526 

30 

.231377 

53 

.068326 

80 

.020177 

6 

.746215 

31 

.?20359 

56 

.065073 

81 

.019216 

^.    7 

.710681 

32 

.209866 

57 

.061974 

82 

.018301 

Wk    8 

.676839 

33 

.199873 

58 

.0:>9023 

83 

.0174^^0 

Wt        9 

.644609 

34 

.190355 

59 

.056212 

84 

.016600 

10 

.613913 

35 

.181290 

60 

.053536 

85 

.013809 

11 

.584679 

36 

.172657 

61 

.050986 

86 

.015056 

12 

.556837 

37 

.164436 

62 

.048558 

87 

.014339 

13 

.530321 

38 

.156605 

63 

.046246 

88 

.013657 

1* 

.505068 

39 

.149148 

64 

.0440t4 

89 

.013006 

15 

.481017 

40 

.142046 

65 

.041946 

90 

.012387 

16 

.45M112 

41 

.135282 

66 

.039949 

91 

.011797 

17 

.436297 

42 

.128840 

67 

.038047 

92 

.011235 

18 

.415521 

43 

.122704 

68 

.036235 

93 

.010700 

19 

.395734 

44 

.116861 

69 

.034509 

94 

.010191 

20 

.376889 

45 

.111297 

70 

.032366 

95 

.009705 

21 

.358942 

46 

.105997 

71 

.031301 

96 

.0'<  9243 

22 

.341850 

47 

.100949 

72 

.029811 

97 

.008803 

23 

.325571 

48 

.096142 

73 

.028391 

98 

.008384 

24 

.310068 

49 

.091564 

74 

.027039 

99 

.007985 

25 

295303 

50 

.087204  75 

.02  5753 

100 

.007604 

In  order  to  find  the  present  worth  of  any  sum  which  is 
to  be  received  at  the  end  of  a  certain  number  of  years. — 
Multiply  the  number  in  the  table  opposite  to  the  term  of 
years,  by  the  sum,  and  the  product  will  be  the  answer, 

Ex.  1.  What  is  the  present  value  of  750^   to  be  re 
ceived  at  the  expiration  of  9  years  ? 


244 


LIFE    ANNUITIES. 


The  number  in  the  table  even  with  9  years  is  .644.609. 
which  IS  to  be  multiplud  by  750. 


.644609 
750 


3223045 
4512263 

483.45675 
20 

y.l350 
12 

1.620 
4 

Answer,  483^  9s.  l^i.  2.48 

Ex.  2.  What  is  the  present  value  of  574^.  10<f.  6(1.,  ta 
be  received  15 years  hence  f  Ans.  276^.  6s.  lid. 

Case  V.  To  find  the  value  of  a  given  sum  payable  at 
the  decease  of  a  person,  whenever  that  sliall  happen. 
That  is,  to  find  the  value  of  an  assurance  of  any  givea 
sum  on  the  whole  duration  of  life. 

Rule.  Subtract  the  value  of  the  lifi  from  the  perpe- 
tuity. Multiply  the  remainder  by  tiie  prodnct  of  the 
given  sum  into  the  rate,  and  this  last  product  divided  by 
100/.  increased  by  its  interest  (^)r  a  year,  will  give  an 
answer  in  a  single  present  payment.  This  payment: 
divided  by  the  value  of  the  life  will  give  the  answer  ia 
annual  payments  during  the  continuance  of  life. 

Ex.  1.  What  ought  I,  who  am  now  45,  to  pay  to  as- 
sure on  my  life  L.lOOO  ;  that  is,  what  ought  I  to  pay  an- 
nually, to  insure  to  my  children  at  my  decease  /..JOOOj 
allowing  money  at  5  per  cent.  ? 


tiFE  ANNUITIES.  245 

The  value  of  alife  of  45,  by  Table,  p.  236,  is  11,105» 
IGO 

and  the  perpetuity  is =  20.  Therefore,  by  the  rule ; 

5 
20—11.105  =  8.895,  which  multiplied  by  5000,    gives 

44475 

441.75  ;  this,  divided  by  105,  or =  423^  1  Is.  Sd., 

105 
equal  the  answer  in  a  single  present  payment.  Therefore 
423/.  Us. 

=  SSl.  2s.  10c?.  nearly,  in  annual  payments 

11,105 
continued  during  life. 

Ex.  2.  Let  the  life  be  20 :  the  sum  L.lOO,  and  the 
rate  5  per  cent.  ? 

The  value  of  a  life  of  30  is,  by  Table,  p.  236,  equal  to 
13.072,  and  the  perpetuity  20.     Therefore,  20  —  13.072 
=  6.928,   which,   niultipled  by  500,  gives  3464,  which 
3464 

divided  by  105,  or =  33^  nearly,  being  the  sum  to 

105 

33 

be  paid  in  a  single  payment ;  and >  =  2l.  lOs.  6d» 

13  072 
nearly,  in  annual  payments  continued  during  life. 

If  tlie  interest  of  money  be  supposed  4  per  cent.,  then 
the  value  of  a  life  of  30  is  equal  14.68,*  and  the  perper- 
100 

tuity  is  equal =  25.      Therefore  25  —  14.68  = 

4 

4123 

10.32.    This  multiplied  bv  400/.  =  4128.     And = 

104 
391'   1 4s. 

39/.  14s.  nearly  ;  and =  2/.   14s. 

14.63 


•  This  is  taken  from  a  table  not  in  this  book.     See  Price'* 
Reversionary  Payments,  and  Morg^an's  Doctrine  of  Annuities,  &tt 
SI* 


246  LIFE    ANNUITIES.- 

Hence  it  appears,  that  when  the  values  are  required 
in  a  single  payment,  the  difference  in  the  rate  per  cent. 
is  considerable,  though  but  trifling  when  made  in  annual 
payments  during  life.  In  this  question,  if  money  be  im- 
proved at  5  per  cent.,  tiie  value  of  the  single  payment 
would  be  33/.  ;  but  at  +  per  cent,  it  would  be  39/.  14s., 
which  is  one  fifth  more  in  the  latter  case  than  in  the  for- 
mer :  l)ut,  when  the  value  is  pai<l  in  annual  sums  during 
life;  at  5  percent.,  each  payment  is  2L  10;>".  6c?.,  and 
at  4  percent,  it  is  2/.  14,';.,  making  a  difference  ot  3s. 
6'/.  per  annum,  being  an  increase  of  less  than  one 
fourteenth. 

If  the  first  of  the  annual  payments  is  to  be  mado. im- 
mediately, then   the  single  payment  is  to  be  divided  by 
the  value  of  the  life,  with  unity  aided  to  it,  so    that  at 
33 

5  per  cewt.  it  will  be =  2/.  6s.  lid,  nearly  5  and 

14-.0r2 
39/.   14s. 

at  4  per  cent,  it  will  be —  =  21.  9s.  4-^d. 

15. OH 
Ex.   3.  Let  the  life  be   25,  the  sum   lOOO/.,  and  the 
rate  5  per  cent.  Answer,     21/.  annually. 

Ex.  4.  Let  the  life  be  60,  the  sum  1 000/.,  and  the 
rate  5  per  cent.  Answer,     39/.  nearly. 

Case  VL  To  determine  the  value  of  an   annuity  cer- 
tain on  a  given  life  for  any  number  of  years. 

Rule.  Find  the  value  of  a  life  as  many  years  older 
than  the  given  life  as  are  equal  to  the  term  for  which  the 
annuitv  is  proposed.  Multiply  this  value  by  1/.  payable 
at  the  end  of  this  term,  and  also  by  the  probability  that 
this  life  will  continue  so  long.  Subtract  the  product 
from  the  present  value  of  the  given  life,  and  the  re- 
mainder multiplied  by  the  annuity  will  be  the  answer. 

Ex.  I.  Let  the  annuity  be  50^  the  age  of  the  given 
life  30  years,  and  the  term  proposed  15  years  ;  interest 
5  per  cent. 


LIFE    ANNUITIES.  24^7 

The  value  of  a  life  of  45,  or  15  years  older  than  the 
given  life,  by  Table,  pa^^e  2)6,  =  11.105.  The  value 
of  1/.  payable  at  the  end  of  15  years  is,  by  table,  page 
243,  IZ  .481  :  and  the  probability  that  the  life  of  30  will 

3^48 

exist  so  long,  is  by  Table,  page  232  = =  .74  near- 

4385 
ly.    Therefore  11.105  X  .481   X  .74  =  3  953.    And  th^ 
present  value  of  the  given  life,  by  the  Table,  page  236, 
=  1.3.072:  therefore  "13.072  —  3.953  =9.119,  and  this 
multiplied  by  50  =  4,35/.   195. 

Had  the  interest  been  only  4  per  cent,  the  value 
would  have  been  about  490/.:  that  is,  in  the  one  case 
455/.  19s.,  and  in  the  other  40O/.,  by  a  person  who 
would  insure  an  annuity  of  50'.  per  annum  for  15  years 
certain,  which  depends  on  the  contingency  of  the  life 
of  a  person  aged  30. 

Ex.  2.  Let  the  annuity  be  40?.  the  age  of  the  given 
life  40,  and'  the  term  proposed  20  years. 

Answer,  402/.   IGs.  \0d. 

Case  VII.  To  find  the  value  of  a  given  sum  paya- 
ble at  the  decease  of  a  person,  should  that  happen  with- 
in a  given  term.  In  other  words  :  What  ought  a  per- 
son to  give  for  haviitg  his  life  assured  to  him  for  a  cer- 
tain term  ? 

Rule.  From  the  value  of  an  annuity  certain  for  the 
given  term,  subtract  the  value  of  the  life  for  the  same 
term,  and  reserve  the  remainer.  Multiply  the  value 
of  l/.  due  at  the  end  of  the  given  term,  by  the  perpe- 
tuity, and  also  by  the  probability  that  the  given  life  shall 
fail  in  the  given  term.  The  product  is  to  be  added  to 
the  reserved  remainder,  and  the  sum  multiplied  by  the 
given  sum  :  this  last  product  divided  by  the  perpetuity 
increased  by  unity,  gives  the  value  in  one  present  pay- 
meat. 


24S  LIFE    ANNUITIES. 

Ex.  I .  A  merchant  at  Liverpool,  aged  30,  expects  to 
realize  a  considerable  property  in  the  next  15  years ;  but 
as  he  may  die  before  he  can  accomplish  his  views,  he  is 
willing;  to  insure  on  his  life,  during  that  period,  the  sum 
of  5000/.,  what  must  he  pay  for  the  same  f 

The  value  of  annuity  certain  for  15  years,  by  Table,  p. 
227,  is  equal  to  10.379  ;  and  by  example,  page  247,  the 
value  of  an  annuity  certain  for  15  years  on  a  life  of  SOiz 
9.119;  therefore  10.379— 9.11 2izl.26zireserved  re- 
mainder. 

The  value  of  1/.  to  be  received  at  the  end  of  15  years? 
by  Table,  page  243,  ZI.481  ;    and  the  probability  that  a 
life  of  30  shall  fail  in  1 5  years,  is 
1142  lOp 

< •z:.26  :*   and  the  perpetuity  is  Z220.     There- 

4385  5 

fore,  .481 X  .26x20  =2.5,  and  this  added  to  the  reserv- 
ed remainder  1.26=3.76,  which  multiplied  by  5000,  the 
given  sum,  and  divided  by  21  (the  perpetuity  increased 
by  unity)  is  equal  895/.  5^.  nearly,  the  value  required  in  a 
single  payment.  That  is,  a  person  of  30  must  give  895/. 
5S.  to  secure  to  his  heirs  5000/.  supposing  he  dies  within 
15  years.  Or  he  must  pay  annually  during  the  15  years, 
if  he  live  so  long,  985/.  5s.  divided  by  9.119,  or  98/.  3s, 
4c?.t  for  *^he  same  security. 

NOTES. 

*  The  probability  of  life's  failing,  is  always  equal  to 
the  probability  of  its  continuing,  subtracted  from  unity. 
Thus  the  probability  of  a  life  of  SO  continuing  15  years, 

3S48 
is  by  table,  p.   232,  zz IZ  .74,  and  the  probahili- 

4385 

3^48     4485— S248        1137 
ty  of  its  failing  1  zz: —  — :^ ^^ ^^  •2^' 

4385         4385  4385 

See  Chances,  p.  230. 

t  The  payments  are  supposed  to  be  made  at  the  end 
of  every  year.  But  in  all  assurances,  the  first  premium 
is  paid  immediately,  and  the  remaining  ones  at  the  be- 


LIFE    ANNUITIES.  S4& 

If  money  can  be  improved  at  4  per  cent.  onl>',  then 
the  sum  to  be  paid  at  once  will  be  929/.  4s.  2(1.,  and  the 
annual  payments  will  be  \0\L  nearly. 

Ex.  2.  If  I  live  7  years,  I  shall  receive  20OOZ.  ;  what 
must  I  give  to  insure  my  life  for  that  period,  being  now 
46  years  of  age  ? 

Ans.   b\L  for  each  annual  payment  for  7  years,  if 
he  live  so  long. 

Case  VIII.  To  explain,  by  examples,  the  mode  of  grant- 
ing annuities  by  the  British  Government  established 
in  the  year  1808. 

[The  following  examples  are  deduced  from  the  tables 
printed  and  circulated  by  -Government,  and  which  may 
be  had,  gratis,  at  the  Office,  Bank  Buildings,  Royal  Ex- 
change, London.] 

Ex.  1.  By  the  tables  it  appears,  that  for  every  100^ 
stock  in  3  the  per  cent,  consolidated  annuities,  will  be 
given  annually  for  life,  to  a  person  of  46  years,  5t.  lis.* 
If,  therefore,  a  person  of  that  age  transfer  \OQOL  stock,  he 
will  receive  an  annuity  for  life  of  55l.  10s.  But  he  will 
receive  interest  30/.  and  keep  his  capital  ^  and  to  insure 
660/.  at  the  Equitable,  or  Royal  Exchange  Offices,  he 
must  pay  rather  more  than  4  per  cent,  j  that  is,  he  must 


NOTES. 

ginning  of  ever  year  after  ;  hence  the  proper  divisor  will 
be  the  value  of  the  life  for  one  year  less  than  the  given 
term  added  to  unity,  or,  in  this  case,  the  value  of  a  life 
for  14  years.  And  generally  :  the  divisor  for  determin- 
ing the  annual  payments  must  he  increased  by  unity, 
whenever  it  is  proposed  that  the  first  payment  should  be 
made  immediately.     See  p.  246. 

*  Supposing   stocks  to  be  at  66,  which  they  are  at 
present* 


250  REVERSIONS. 

pay  between  26  and  271.  annually,  during  life,  to  insure 
to  his  heirs  at  his  death  the  660/.  which  he  transfers  to 
Gi)vernment :  he  will  ©f  course  be  a  loser  by  the  transfer, 
of  between  one  and  two  pounds  per  annum.  It  is  there- 
fore obvious,  that  no  one,  when  stocks  are  at  66,  can  join 
in  the  plan  held  out  by  Government,  who  is  not  willing 
to  give  up  his  capital. 

Ex.  2.  When  stocks  are  at  GO,  he  will  receive  for 
lOOO^.  stock,  59.1.  lOs.  ;  and  to  insure  000/.  must  pay 
more  than  24/.  to  insure  his  life,  and  will  of  course  be  a 
loser  of  1/.  lOs.  per  annum. 

Ex.  3.  When  stocks  are  at  80,  as  they  may  be, he  will 
receive  for  the  1  GOO/,  stock  62/.  ;  but,  to  insure  800/. 
he  niu^t  pay  annually  rather  more  than  32/.  ;  in  this  case 
there  will  be  his  interest  left,  and  he  will  he  neither 
gainer  nor  loser. 

These  examples  will  suffice  for  the  whole. 


REVERSIONS. 


Reversions,  or  Reversionary  Annuities,  are  those 
which  do  not  commence  till  after  a  certain  number  of 
years,  or  till  the  decease  of  a  person,  or  some  other  future 
event  has  happened. 

Case  I.  To  find  the  present  value  of  an  annuity  for  a 
term  of  years,  which  is  not  to  commence  till  the  expi- 
ration of  a  certain  period. 

Rule.  Subtract  from  the  value  of  an  annuity  for  the 
whole  period,  the  value  of  an  annuity  to  the  time  when 
the  reversionary  annuity  is  to  commence. 


HEVERSIONS  251 

Ex.  1-  What  is  the  present  value,  at  5  per  cent,  com- 
lund  interest,  of  80/.  per  annum  for  24  years,  com- 
encing  at  the  end  of  8  years  ?  ^4i-\~S=32, 


pound 


The  present  value  of  an  annuity  (Table,  p.  227,)  for 
32  years,  is  15.802677,  and  the  value  of  one  for  8  years 
is  0.4632 1 3,  therefore, 


15.802677 
6.463213 


9.339464  X80=74r.  157 12=747^.  3s.  l|d/. 

Ex.  2.  "What  is  the  present  value  of  an  annuity  of  55?, 
for  15  years,  to  commence  at  the  end  of  15  years  f 

Answer,  274/,  12s. 

Ex.  3.  What  is  the  present  value  of  an  annuity  for  49 
years,  to  commence  at  the  end  of  47  years  ? 

Answer,    Something  more  than  a  year  and  half's 
purchase. 

Case  II.     To  find  the  value  of  an  annuity  certain  for  a 
given  term,  after  the  extinction  of  any  life  or  lives. 

Rule.  Subtract  the  value  of  the  life  or  lives  from  the 
perpetuity,*  and  reserve  the  remainder.  Then  say,  as 
the  perpetuity  is  to  the  present  value  of  the  annuity  cer- 
tain, so  is  the  reserved  remainder,  to  the  number  of  years 
purchase  required. 


NOTES. 

*  Perpetuity,  is  the  number  of  years  purchase  to  be 
given  for  an  annuity  which  is  to  continue  for  ever  ;  and 
It  is  foujid  by  dividing  100/.  by  the  rate  of  interest  j  thus, 


252  HEYERSIONS. 

Ex.  L  What  is  the  value  of  an  annuity  certain  for  14 
y^ars,  to  commence  at  the  death  of  a  person  aged  35,  al- 
lowing 5  per  cent.  ? 

The  value  of  a  life  of  35  (Table,  p.  236)  «=  I2.50g; 
this  subtracted  from  20,  the  perpetuity,  leaves  7.498  = 
reserved  remainder.  Then,  as  20;  9.898t : :  7.498: 
^.7107  =  number  of  years  purchase. 

NOTES. 

allowing  5  per  cent.,  the  perpetuity  is  20  years,  or 
—  =  20  ;  and  at  the  rates  most  usually  adopted,  the 
perpetuity  is  as  follows  : 

At  3  per  cent— r- =  33.33,  &c. 


3i  ditto 

^/°=28.5r,&c. 

4    ditto 

T='^- 

4}  ditto 

—-=  22.22,  &c. 

5    ditto 

100 
5     =2^- 

6    ditto 

100        ,^  ^^    „ 
—T-=  16.66,  &c, 

7    ditto 

7=14.2S,&C, 

8    ditto 

--,2.5. 

These  are  the  number  of  years  purchase  to  be  given 
for  a  perpetual  annuity,  on  the  supposition  that  it  is  re- 
ceivable yearly  :  but,  as  annuities  are  more  commonly 
received  half-yearly,  and  the  interest  of  money  likewise 
paid  half-yearly ;  in  this  case  the  perpetuity  will  be 
somewhat  greater  or  less  than  the  above,  as  the  periods 
at  which  the  annuity  is  payable  are  more  or  less  frequent 
than  those  at  which  the  rate  of  interest  is  here  supposed 
payable. 

t  The  value  of  an  annuity  certain  for  14  years.  Table- 


REVERSIONS.  253 

Ex.  2.  A  and  his  heirs  are  entitled  to  an  annuity  of 
Zr.lOO  certain  for  25  years,  to  commence  at  the  death  of 
a  cousin  aged  43  years  ;  what  can  A  sell  his  interest  in 
this  annuity  for  ?        ^  Answer,  6261.  I6s. 

Case  III-  To  find  the  value  of  an  annuity  for  a  term 
certain  ;  and  also  for  what  may  happen  to  remain  of  a 
given  life  after  the  expiration  of  this  term: 

Rule.  Find  the  value  of  a  life  as  many  years  older 
than  the  given  life,  as  are  equal  to  the  term  for  which  the 
annuity  certain  is  proposed.  Multiply  this  value  by  il. 
payable  at  the  end  of  the  given  term,  and  also  by  tho 
probability  that  the  given  life  will  continue  so  long.  Add 
the  product  to  the  value  of  the  annuity  certain  for  the 
given  term,  and  the  sum  will  be  the  answer. 

Ex»  1.  What  is  the  value  of  an  annuity  of  60/.  for  14 
years,  and  also  for  the  remainder  of  a  life  now  aged  35, 
after  the  expiration  of  that  term  ?         35  +  14  =  49. 
The  value  of  a  life  aged  49  ( I'able  1, 

page  236.) =10.443 

The  value  of  ll.  payable  at  the  end  of 

14  years  (  Fable,  page  243  )       -     -      ==      .50.5068 
The  probability  that  the  life  wdl  exist  >    _2936 
so  long,  (Table,  page  232.)      -         J    "ioib 
2936 

Therefore,  10,443  x  .505063  X =386 1  ;  this  added 

4010 
to  9.898,  the  value  of  an  annuity  certain  for   14  year?, 
(see    Table,  page  227.)  :z:  13  759,  the  number  vS  years 
purchase  ;  and  13.739  X  60  —  825/.  10.>-.  9-^. 

Ex.  2.  What  is  the  value  of  an  annuity  of  73/.  for  10 
years,  and  also  the  remainder  of  a  life  now  aged  24, 
After  the  expiration  of  that  term  ?  Ans.  1070/.  5s. 

Case  IV.  To  find  what  annuity  can  be  purchased  for  a 
given  sum,  during  the  joint  lives  of  two  persons  of 
given  ages,  and  also  during  the  life  of  the  survivor,  on 
condition  that  the  annuity  shall  be  reduced  one-iialf  at 
the  extinction  of  the  joint  lives. 
22 


254  REVERSIONS. 

Rulp:.  Divide  twice  the  given  sum  by  the  sum  of  the 
value  of  the  two  single  lives,  and  the  quotient  will  give 
the  annuity  to  be  paid  during  the  joint  lives,  one-half  of 
which  is  therefore  the  annuity  to  be  paid  during  the 
remainder  of  the  surviving  life. 

Ex.  1.  A  man  and  his  wife,  aged  G5  and  27,  are  de- 
sirous of  sinking  2000/.  in  order  to  receive  an  annuity 
during  their  joint  lives,  and  also  another  annuity  of  half 
the  value  during  the  remainder  of  the  surviving  life : 
what  annuities  ought  to  be  granted  them  ^ 

The  value  of  a  life  of    27  >  .j,  ,  j     .         335  S  ='3.377 
The 35  S  ^^^^  ^'  P-  ^"^^  ?  =12.502 


^5.879 
4000  (twice  the  sum) 

Therefore,  « 1=154^.   lis.  Sc/.  =: 

25.879 
annuity  during  their  joint  lives:  and   77^  5s.  l-^d,  an- 
nuity during  the  Hie  of  the  survivor. 

Ex.  2.  A  single  man,  aged  60,  possessed  of  1500/.  is 
tiesirous  of  purchasing  with  it  an  annuity  for  himself  and 
his  sister,  aged  40,  during  their  joint  lives,  with  one 
of  half  the  value,  during  the  remainder  of  the  life  of  the 
survivor,  at  the  death  of  either  :  what  will  be  the  value 
of  the  annuities  ? 

Answer,  148/.  6s.  annuity  during  joint  lives,  and 
74/,  Ss,       do      for  the  survivor. 

Ex.  3.  A  man  possessed  of  lOOO/.  which  he  will  sink 

in  the  same  way,  and  for  the  same  purposes,  during  the 

joint  lives  of  himself  and  father  ;  the  age  of  the   one  is 

65,  of  the  other  80  :  what  annuities  can  be  given  for  it  ? 

Answer,   155/.  annuity  during  joint  lives,  and 

77/.  10s.     do     for  thesurvivor. 

V.  To  find  the  value  of  the  expectation  of  a  perpe- 
tual annuity,  provided  one  person  of  a  given  age  sur- 
vives another  of  a  given  age. 


r 


REVEUSIONS.  S55 

(l.)  If  the  Expectant  be  the  elder* 

Rule.  Find  the  value  of  an  annuity  on  two  equal  joint 
lives,  whose  common  a^^e  is  equal  to  the  age  of  the 
oldest  of  the  two  proposed  lives  ;  subtract  this  value 
from  the  perpetuity,  and  take  half  the  remainder  :  tlieii 
say, 
As  the  expectation  of  the  duration  of  life  of  the  younger, 

Is  to  that  of  the  elder  : 

So  is  the  iialf  remainder  to  a  fourth  proportional  : 
which  will  be  the  number  of  years  purchase,  if  the  ex- 
pectant is  the  older. 

(2.)  If  the  Expectant  be  the  younger. 

Add  the  value  found,  as  above,  to  that  of  the  joint 
lives,  and  let  the  sum  be  subtracted  from  the  perpetuity, 
and  the  remainder  is  the  answer. 

Ex.  1.  What  is  the  value  of  B's  expectation,  (aged  SO,) 
of  an  estate  50/.  per  annum,  provided  he  survive  A 
aged  20  ? 

Value  of  two  joint  lives,  aged  30,  (Table  TI.  p.  239) 
=  10.255,  the  dirterence  between  which  and  20,  (the 
perpetuity,)  is  9.745,  the  half  of  which  is  4.872  :  there- 
fore, 

:  :  4  872  :   4.119  ZZ  205/.   I9s. 

Ex.  2.  What  is  tlie  value  as  above,  when  B  is  20^ 
and  A  30  .^ 

Then,  to  4.1 1 9,  just  found,  add  [p.  239) 

10.707,  value  of  the  joint  lives    (Table  11. 

14.826;  this  subtracted  from  20,  the  per- 
petuity and  the  remainder,  5-174  x  50  =  258/,  14^.  is 
the  uae  answer. 


^^^  ~      REVERSIONS. 


^      EXAMPLES    FOR     PRACTICE. 


Ex.  1.  What  is  the  difference  in  the  value  of  an  an- 
nuitj  of  20/.  certain  for  30  years,  and  an  annuity  of  the 
same  amount  on  the  longest  of  two  lives,  ao;ed  25  and  40  ? 
Answer,  L.5  4  4|  difference. 
Ex.  2.  What  is  the  valu6  of  an  estate  of  150/.  per 
annum  held  on  the  longest  of  two  lives,  ased  40  and  50, 
suhject  to  the  payment  of  an  annuity  of  14/.  to  a  life  of 
62,  and  another  annuity  of  18/.  to  a  life  of  €5  } 

Answer,    1847/.  16.«.  value 
Ex.  .S.  What  is    the  present  worth    of  2000/.   to  be 
received  at  the  decease  of  a  person  aged  65  } 

Answer,  1272/.  8^-.  present  worth. 
Ex.  4.  W^hat  is  the  present  value  of  36/.  a  year,  being 
the  third  part  of  a  farm  in  Essex,  after  the  death  of  a 
person  aged  54  years  .? 

Answer,  375/.   11. «J.  9^^.  present  value. 
Ex.  5.  What  is  the  present  value  of  a   reversionary- 
annuity  of  252/.  S.«.  8c/.  during  the  life  of  a  person  aged 
24,  in  oase  he  survives  his  brother,  a2;ed  34  ? 

Answer,   1539/.  5s.  9d.  present  value. 

Ex  6.  What  shouJd  be  the  consideration  to  be  paid  at 
the  death  of  a  person  aged  85,  for  1000/.  now  advanced 
to  a  person  aged  25,  in  case  the  latter  survives  the  for- 
mer ?  Answer,  1193/.  tis. 

Ex.  7.  What  is  the  value  of  the  reversion  of  PI/,  per 
annum  forever,  after  the  death  of  a  person  aged  53  ? 

Answer,  932/,  18.v.  Id,  value; 
Ex.  8.  A  person  aged  52,  is  entitled  to  800/.  at  the 
death  of  another  aged  76,  provided  the  former  survives 
the  latter ;  what  is  its  present  worth  ? 

Answer,  522/.  Os.  9d, 

Ex.  9.  What  is  the  present  value  of  an  annuity  on 
the  longest  of  two  lives,  now  aged  25  and  30,  the  an- 
nuity not  to  commence  till  14  years  hence  ? 

Answer,  854/.  19s.   Id. 


LEASES. 


LEASES. 


257 


A  Lease  is  a  conveyance  of  any  lands  and  tenements, 
made,  in  consideration  of  rent,  or  of  a  present  sum  of 
money,  for  life,  or  for  a  term  of  years. 

The  purchaser  of  a  Lease  may  be  considered  as  the 
p^irchaser  of  an  annuity  equal  to  the  rack-rent  of  the 
estate;  its  value  must  therefore  be  calculated  on  the 
same  principles  as  that  of  an  annuity. 

The  sum  paid  down  for  the  grant  of  a  lease  is  so  much, 
as  being  put  out  to  interest  will  enable  the  landlord  to 
repay  himself  the  rack-rent  of  the  estate,  or  the  year- 
ly'value  of  his  interest  tiierein. 

The  value  of  the  lease  depends  on  the  length  of  the 
term,  and  the  rate  of  interest  which  the  landlord  can 
make  of  his  money. 

The  value  of  leases  at  5  per  cent,  compound  interest 
may  be  found  in  the  Table  page  227. 

Thus,  the  value  of  a  lease  for  14  years,  of  a  farm 
worth  150^  per  annum,  is  by  that  table,  9.898641X150 
=  1484^   \5s>   l\d.    - 

Kx.  I.  What  ought  to  be  given  for  a  lease  of  26  years 
of  an  estate  of  18^  per  annum  clear  annual  rent,  in-or- 
<ler  that  a  purchaser  may  make  5  per  cent  of* his  tn  )nf»y  ? 

Aijswer  238^.   15s. 

Ex.  2.  A  friend  has  just  purchased  the  lease  of  a 
house  fgr  54  years,  for  which  he  gave  S.'Ol.  and  he  is  to 
pay  a  ground-rent  of  \l  per  annum :  how  much  ought 
the  house  to  let  for,  allowing  5  per  cent,  interest  only  ? 

Answer  30^  12s.  4d. 

lieases  are  generally  calculated  at  a  higher  rate  of  in- 
terest; we  shall  therefore  insert  the^'ollowino; 

TABLK,. 
Shewing  the  Number  of  Y^ars  Purchase  that  ought  to  be 
,..  given  for  a  Lease,  for  any  Number  of  Years  not  ex» 

ceeding  100,  at  6,  7,  and' 8  per  cent,  interest. 

2:2* 


258 


LEASESi 


Yrs. 

6  per  cent. 

7  percent. 

8  per  cent. 

1 
Year. 

6  per  cent. 

7  per  cent. 
13^324' 

8  per  cent. 

"~T 

.94:53 

.9345 

,9259 

51 

15,8130 

12,2532 

2 

1.8333 

l,8u80 

1,7832 

52 

15,8613 

13,8621 

12,2715 

3 

26730 

2,6243 

2,5770 

53 

15,9009 

13,8898 

12,2884 

4 

3.4651 

3,3872 

3,.3l2l 

54 

15,9499 

13,9157 

12,3041 

5 

4.2123 

4,l'K)l 

3,9927 

55 

15,9905 

13,9399 

12,^186 

6 

49173 

4,7665 

4,6228 

56 

16,0288 

13,9265 

12,3320 

7 

5.582.S 

5,3892 

5,2063 

57 

16,0649 

13,9837 

12,3444 

8 

6  2097 

5,9712 

5,7466 

58 

16,(t989 

14,0034 

1    ,.;560 

9 

6  8016 

6,5 1 J2 

6,2468 

59 

16,1311 

14.02 1'j 

12,.%69 

10 

7.36(JO 

7,0235 

6,7100 

60 

16,1614 

14  0.391 

12,3765 

11 

7  8868 

7,4986 

7,1389 

61 

16,1900 

14,0553 

l'J,3856 

12 

8  3838 

7,9426 

7,5360 

62 

16,2170 

14,0703 

12,3941 

l.i 

8.8.526 

8,.>576 

7,' 037 

63 

16,2424 

14,0844 

12,4020 

14 

9  2919 

8,7454 

8,2442 

64 

16,2664 

14,09/6 

12,4092 

15 

9.7122 

9,l(»79 

8,5594 

65 

16,2891 

i  4. 1099 

12,4159 

16 

UK  105 8 

9,44436  1 

8,8513 

66 

16  3104 

14,1214 

12,4222 

17 

10  4772 

9.76.32 

9,1216 

67 

16,3306 

14,;  321 

12,4279 

18 

10.8276 

10,0590 

9,3718 

68 

16,3496 

14,1422 

12,4333 

ll> 

11  1581 

10,3.S55 

9,6035 

69 

16,3676 

14,1516 

12,4382 

fiO 

1 1  4099 

10,5940 

9,8181 

70 

16,3845 

14,1603 

12,4i28 

iil 

11764(^ 

10,8355 

10,0168 

71 

16,4005 

14,1685 

12,4470 

22 

12.0415 

11,0612 

10,2007 

72 

16,4155 

14,1762 

12,4509 

i.l 

12  3033 

11,2721 

10,3710 

73 

16,4297 

14,1834 

12,454r» 

'24 

12  5503 

11,469.) 

10,5287 

74 

16,4431 

14,1901 

12,4579 

25 

127S.n3 

11,6535 

10,6747  1 

75 

16,45  5  S 

14,1963 

12,4610 

■26 

13  00,31 

11,8257 

10,8099  1 

76 

16,4677 

14,2022 

12,4639 

'27 

13.2W3 

11,9867 

10,9351   • 

77 

16,4790 

14,2076 

12,4666 

28 

13  44)61 

12.1371 

11,0510 

78 

16,4896 

14,2127 

12,4691 

39 

13.59U7 

12.2776 

11,T584 

79 

16,4996 

14  2175 

12,4713 

30 

13  76  i  8 

12,40110 

11,2577 

80 

16,5091 

14,2220 

12,4735 

31 

1 3  9290 

12,5318 

11,3497 

81 

16,5180 

14,2261 

12,4754 

3/ 

14  0810 

12/i465 

11,4349 

82 

16,5264 

14,2300 

12,4772 

33- 

14  2.302 

12,7537 

11,5138 

83 

16,5343 

14.2337 

12,4789 

;i4 

r  •  .3681 

.12,8540 

11,5869 

84 

16,5418 

14,2371 

12,48(5 

55 

144982 

12,9476 

11,6545 

85 

16,5489 

14,2402 

12,4819 

36 

14  6209 

13,0352 

11,7171 

86 

16,5556 

14,243: 

12.4833 

;37 

147367 

13,1170 

a. 7751 

87 

16,5618 

14,2460 

12,4845 

^8 

14  8460 

13,1935 

ll,8J88 

88 

16,5678 

14,2486 

12,4856 

->9 

14  y  UiO 

13,2649 

11,8785 

89 

16,5734 

14,2510 

12,4867 

4o 

15.0  Mri 

13.3317 

11,9246 

QO 

16,5787 

14,2533 

12,4877 

4! 

15.1.38'' 

13,3941 

11,9672 

91 

16,5856 

14,2554 

» 2,4886 

4^i 

15.2245 

1.).4524 

12,00r.6 

92 

16,5883 

14,2574 

12,4894 

4.S 

15  3061 

13,5069 

12,0432 

93 

16.5928 

14,2.S92 

12,4902 

44 

15  3831 

13,5579 

12,0770 

94 

16,5969 

14,2610 

12,4905 

45 

15  4558 

13,6055 

12,1084 

95 

16,6009 

1 4,2626 

12  411 G 

46 

15.524.3 

i  13,6500 

1 -•,1.374 

96 

16,6046 

14,2641 

12,4922 

47 

15  5890 

13,6916 

12.1642 

97 

lft,6081 

14,2655 

12,4928 

'  48 

15.6500 

13,7304 

12,1891 

98 

16,6114 

14,2668 

12,4933 

49 

15  7075 

13,7667 

12,2121 

99 

16,6145 

14,2680 

12,4938 

50 

15  7618 

13,8007 

12,2334 

100 

16,6175 

14,2692  'r'.4943 

XEASES.  259 


Case  1.  To  find  the  sum  that  oiuht  to  be  given  fo 


lease. 


Rule.  Look  in  the  table  an;ainst  the  number  of  years 
for  which  th»!  leu^e  is  to  continue,  and  on  the  line  even 
with  it,  under  the  given  rate  of  interest,  is  the  number 
of  years  purchase  that  ou^ht  to  be  given  for  the  same. 

Ex.  What  sum  ought  to  be  given  for  the  lease  of  an 
estite  of  17  years,  of  the  clear  annual  rent  of  75i.  al- 
lowing the  purchaser  to  make  7  per  cent,  interest  of  his 
money  ? 

Answer,  9.7632x75— 73 2.24=732^.  4s.  9|^, 

T*Tx.  2.  What  must  he  given  for  a  lease  of  21  years, 
at  the  clear  annual  rent  of  50  guineas,  allowing  8  per 
cent,  for  money  ?  Answer,  525L  n."-.  9d, 

Ex.  3.  What  is  the  worth  of  a  lease  of  83  years  of  aH 
estate  of  78/.  per  annum,  interest  being  6  per  cent..^ 

Answer,  1239/.  13s. 

Ex.  3.  What  sum  ought  to  be  given  for  a  lease  of  69 
years,  of  aTarm  of  150^  per  annum,  the  purcliaser  being 
alioAed  6  percent,  for  his  money  ?     Ans.  24i55L  \0s. 

Ex.  5.  What  sum  ought  to  be  given  for  the  lease  of 
46  years,  of  an  estate  estimated  at  200^,  but  which  is 
charged  with  the  payment  of  a  reserved  rent  of  70/.  i5s. 
besides  taxv^s  and  incidental  expenses  to  the  amount  of 
49/.  l^s.  annually;  allowing  tl»e  purchaser  6  per  cent, 
interest  for  his  money  .^  Answer,   1236/.  9s.  9rf. 

Ex.  6.  What  sum  ought  to  be  given  for  the  ground 
rent  of  a  house  of  15/.  per  annum,  for  18  years,  allow- 
ing the  purchaser  8  per  cent  .'^        An«.   140/.   lis.  6c?. 
«. 

Cask  II.  To  find  the  annual  rent  correispondirig  to  any- 
given  sum  paid  for  a  lease. 

Rule.  Divide  the  aura  paid  for  the  lease  by  the  num- 
ber of  years  purchase  that  are  found  against'  the  given 
term,  and  under  the  rate  of  interest  intended  to  be  made 
of  the  purchase  money,  the  quotient  will  be  the  anauai 
rent  required". 


260  LEASES. 

Kx.  I.  I  a<m  asked  1500/.  for  a  40  years  lease,  to 
what  annual  rent  is  that  equivalent,  allowing  6  per  cent 
for  money  ? 

1500 

"  Answer, =99/.  1 3s.  lie?,  nearlv. 

15.046 

Ex.  7.  If  I  sell  the  lease  of  my  house,  which  lias  81 
years  to  ran,  for  800  guineas,  at  what  rent  will  the  pur- 
chaser stand,  who  will  have  a  ground  rent  of  5L  5s.  per 
annum  to  pay  likewise,  allowing  7  per  cent? 

Answer,  64/    5s. 

Case  III.  To  find  the   number  of  years  purchase  given 
for  a  lease  that  cost  a  certain  sum  of  money. 

Rule.  Divide  the  sum  paid  for  the  lease  by  the  clear 
annual  rent  of  the  estate  for  vvhicli  i-  is  given,  and  the 
quotient  will  be  the  number  of  years  purchase  required. 

Ex.  1.  The  lease  of  a  house,  at  the  clear  annual  rent 
of  1 16/.  was  sold  for  1630/.,  what  number  of  years  pur- 
chase was  given  for  it.'' 
1630 

ZZ14  vears,  0  months,  2  weeks,  4  daySo 

116 

Ex.  2.  How  many  years  purchase  did  the  lease  of  a^ 
house  sell  for  which  cost  800/.  and  the  rent  was  60  gui- 
neas? Answer,  12  years,  8  month,  12  daj'S. 


FREEHOLDS 


Case  I.  To  find  the  gross  sum  which  ought  to  be  paid 
for  a  freehold  estate. 

Rule,  (l)  "Multiply  the  number  of  years  purchase 
by  the  annual  rent.'^  Or,  (2)  "  Multiply  the  annual 
rent  by  100,  arid  divide  the  product  by  the  rate  of  inte- 
rest which  it  is  proposed  to  make  of  money ;  the  quo- 
tient will  be  the  sum  required." 


FREEHOLDS.  261 

Kx»  What  oui^lit  I  to  give  for  a  freehold,  the  rent  of 
which  is  751.  per  annum,  supposing  I  mean  to  make  4 
per  cent,  of  [i\y  money  ? 

\jy  the  1st.  Uule,  "the  answer  is  25  X  75    =  1873/. 

75  X  100 

By  the  2d.  —    .—    —    -, =1875/. 

4 
If  I  had  wanted  5  per  cent,  for  my  money,  the  answer 
would  have  been     -     l^t.  20  x  75    =  1600/. 
75  X  100 
2d.      .,    .      .      ==  150O/. 
5 

But  if  I  were  contented  with  3  per  cent.,  then  I  might 
aftbrd  to  give  for  it  2500/. 

75  X  100 
=  2500?. 


Case  TI.  To  find  the  clear  annual  rent  which  a  freehold 
ought  to  produce,  so  as  to  allow  the  purchaser  a  giveia 
rate  of  interest  for  his  money  ? 

Rule.  Multiply  the  sum  paid  for  the  same,  by  the 
given  rate  per  cent.,  and  divide  by  100,  the  quotient  will 
be  the  annual  rent  required. 

Ex.  A  person  has  given  3000  guineas  for  a  freehold - 
estate,  and  wishes  to  let  it  so  as  to  have  4i  per  cent,  for 
his  money,  what  must  be  the  annual  rent  ? 

Answer,  l4l/.  I5s. 

Case  III.    To  find  the  value  of  a  freehold,  to  be  entered 
upon  after  a  certain  term. 

Rule.  Subtract  the  value  of  that  certain  term,  from 
the  value  of  the  perpetuity,  and  the  diftereaee  will  be 
the  true  value. 

Ex.  1.  "W'batsum  should  be  given  for  the  reversion  of 
a  freehold  after  14  years,  allowing  interest  6  per  cent^ 
and  the  clear  annual  rent  85/. 


262 


RENEWAL    OF    LEASES. 


Value  of  a  lease  of  14  years,  Table,  page  258,  =  9.295  ; 
which  subtracted  from  "16.667,  the  perpetuity,  leaves 
'2. 372;  and  fhis  multplied  by  85/.  gives  tiie  value  = 
626/.    12s.  4jf/. 

Kx.  2.   Wliat  ought  I  to  give  for  the  reversion   of  a 
freehold  worth  120/.  per  annum  ;  but  a  lease  of  which  is 
sold  for  5  years  t©  come,  supposing  interest  5  p^r  cent. 
Answer,   1880/.  10s.  5d.  nearly. 


RENEWAL  OF  LEASES. 


Case  T.  To  ascertain  what  fine  should  be  given  for  the 
renewal  of  any  number  of  years  lapsed  in  a  lease  ori- 
ginally granted  for  34  years. 

Rule.    This  is  done  by  means  of  the  following 
TABLE, 
For  Renewing  any  Number  of  Years  lapsed  in  a  lease  for 
Twenty-one  Years. 


i.11.564 

Years. 

3  per  Ct. 

4  per  Ct. 

5  jjer  Ct. 

0  per  Ct. 

3  per  Ct. 

per  Cent. 

1 

,538 

,439 

,359 

,294 

,199 

,100 

2 

1,091 

,895 

,736 

,606 

,413 

,213 

3 

1,661 

1,870 

1,132 

,936 

,645 

,338 

4 

2,249 

1,863 

1,547 

1,287 

,895 

,477 

5 

2,854 

2,377 

1,983 

1,658 

1,165 

,633 

6 

3,477 

2,911 

2,441 

2,052 

1,457 

,806 

7 

4,119 

3,466 

2,922 

2,469 

1,773 

1,000 

8 

4,780 

4,043 

3.428 

2,911 

2,113 

1,216 

9 

5,461 

4,644 

3;958 

3,380 

2,481 

1 ,457 

10 

6,162 

5,269 

4,515 

3,877 

2,878 

1,726 

11 

6,885 

5,918 

5,099 

4,404 

3,507 

2,026 

12 

7,629 

6,594 

5,713 

4,962 

3,770 

2,361 

13 

8,395 

7,296 

6,358 

5,554 

4,270 

2,734 

14 

9,185 

8,G2r 

7,035 

6,182 

4,810 

3,151 

15 

9,998 

8,787 

7,745 

6,847 

5,394 

.'^,616 

16 

10,835 

9,577 

8,49i 

7,552 

6,024 

4,135 

17 

1 1,698 

10,399 

9,275 

8,299 

6,705 

4,713 

18 

12,586 

11,254 

10,098 

9,091 

7,440 

5,3^9 

19 

13,502 

12,113 

10,962 

9,931 

8,234 

6,079 

30 

14,444 

13,068 

11,869 

10,821 

9,091 

6,882 

total 

15,415 

14,029 

12,821 

11,764 

10,017 

7,779 

RENEWAL   OF    LEASES.  2t)3 

Ex.  1.  What  ought  to  be  given  as  a  fine  for  the  renew- 
al of  15  years  lapsed,  or  expired  in  a  lease  for  21  years, 
allovving  the  tenants  percent,  interest,  and  estimating 
the  clear  and  improved  rent  at  60  guineas  per  annnm  ? 

Against  15  in  the  table,  and  under  5  percent.,  is  7.745, 
and  this  multiplied  by  63/.  gives  487.935  =  487/.  I85. 
Sid, 

If  the  interest  agreed  on  had  been  6  or  8  per  cent.,  the 
answers  would  have  been 

6.847  X  63  =  431/.  7s.  2^. 
Or,  5.394  X  63  =  339/  16^.  5d, 

Ex,  2.  What  ought  to  be  given  to  a  landlord  for  add- 
ing seven  years  to  a  lease,  of  which  fourteen  years  are 
unexpired,  allowing  the  tenant  6  per  cent  interest  for 
his  money,  and  the  improved  rent  to  be  60/.  per  annum  ? 

Answer,  148/.  2s.  9ld. 

Case  II.  To  ascertain  the  value  of  the  fine  which  ought 
to  be  paid  for  renewing  a  given  number  of  years  in  any 
lease. 

HuLE.  The  value  for  renewing  an  additional  term, 
or  for  adding  any  number  of  years  to  the  unexpired  part 
of  an  old  lease,  is  equal  to  the  difference  between  the 
value  of  the  lease  for  the  whole  term,  and  the  value  of 
the  unexpired  part. 

Ex.  1.  What  ought  to  be  ^iven  for  the  addition  of 
seven  years  to  a  lease,  of  which  13  are  unexpired;  al- 
lowing 6  per  cent,  for  money  ? 

The  whole  term  for  which  the  new  lease  is  to  be 
granted  is  20  years  ;  therefore,  Table,  under  6  per  cent, 
and 

against  20  is  1 1 .469,  and 

against  13  is  8.852;  therefore  this  last  subtracted 
from  the  former  will  leave  2.617  for  the  number  of  years' 
purchase  which  ought  to  be  given  for  the  renewal. 

Ex.  2.  What  should  be  given  for  the  completing  a  60 
years*  lease,  of  which  a  tenant  has  an  unexpired  term  of 
15  years,  allowing  him  7  per  cent,  for  his  money  > 

Answer,  4.932  year's  rent 


9.64f  RENEWAL    OF    LEASES. 

Ex.  S.  I  have  a  house  for  a  lease  of  48  years,  but  I 
wish  to  extend  the  lease  to  97  years  :  how  much  must  1 
pay  for  it,  supposing  the  house  worth  50^.  per  annum, 
and  the  interest  8  per  cent.?  Ans.  15/.  4s. 

It  will  be  seen  by  working  Ey.  2.  of  Case  1,  by  this 
rule,  that  the  answer  will  be  precisely  the  same  by  both 
metliods  :  for  the  whole  term  for  which  the  new  lease  is 
granted  is  21  years  :  the  value  of  a  lease  for  this  term  is, 
by  Table,  1 1.764,  and  the  value  of  the  14  years'  lease  yet 
to  come  is  9.295;  this  subtracted  from  the  other,  gives 
2.4G9,  as  before,  which,  multiplied  by  60,  and  the  an- 
swer is  \4SL  2s.  did. 

The  following  table  will  comprehend  the  cases  that 
most  frequently  occur  at  ti>e  rate  of  5  and  6  per  cent. 


RENEWAL    OF    LEASES. 


265 


TABLE, 

For  Renewing,  with  one  Life,  the  Lease  of  an  Estate  held 
on  Three  Lives. 


» 


Life 

Age  of 

1               1 

Life 

Age  of 

put 

lives  in 

5pr.Ct.  6pr.Ct.| 

put 

lives  in 

5pr.  Ct 

6pr.Ct, 

in. 

[)ossession. 

•1,741 

in. 

possession. 

30-30 

1,305 

40—75 

3,943 

3.076 

30-40 

2,035 

1,521 

50-50 

3,289 

2,536 

30-50 

2,431 

1,832 

50—60  \  3,910 

3,059 

30—60 

2,838 

2,160 

50-70 

4,546 

3,579 

30—70 

3,277 

2,535 

15 

50-75 

4.816 

3,819 

30—75 

3,402 

2,571 

60-60 

4,692 

3,678 

40-40 

V,397 

1,792 

60-70 

5,780 

4,327 

40-50 

2,916 

2,204 

60-75 

6,054 

4,849 

40_60 

3,451 

2,637 

70-70 

7,125 

5,805 

10 

40_70 
40_  75 
50_50 
50_60 
50—70 
50-75 
60—60 
60_70 
60—75 
7C-70 

3,914 
4,264 
3,563 
4,206 
4,873 
5,174 
5,023 
6,161 
6,452 
7,556 

3,03  J 
3,273 
2,723 
3,242 
3,819 
4,062 
3,911 
4,917 
5,U2 
6,124 
1 

30-30 
30-40 
30—50 
30-60 
30—70 
30-75 
40-40 
40-50 
40-60 

1,404 
1,673 
2019 
2,363 
2,813 
2,845 
2,027 
2,467 
2,943 

1,079 
1,284 
1,557 
1,831 
2,213 
2,241 
1,558 
1,908 
2,293 

" 

20 

40—70 

3,358 

2,641 

30-30 

1,572 

1,191 

40-75 

3,615 

2,873 

30-40 

1,857 

1,407 

50-50 

3,010 

2,841 

30—50 

2,227 

1,699 

50—60 

9,607 

2,828 

30-60 

2,600 

1,996 

50-70 

4,208 

3,337 

15 

30-70 

3,052 

2,381 

50—75 

4,474 

3,576 

30-75 

3,127  1  2,403 

60-60 

4,347 

3,433 

40—40 

2,224 

1,687 

60-70 

5,386 

4,338 

40    50 

2,701 

2,067 

60-75 

5,636 

4,558 

40-60 

3,205 

2,474 

70-70 

6,695 

5,489 

40-70 

3,641 

2,839 

Rule.  The  years'  purchase  in  the  tablej  multiplied  by 
the  improved  annual  value  of  the  estate,  beyond  the  rent 
payable  under  the  lease,  gives  the  fine  to  be  paid  for  put- 
ting in  the  new  life. 

S3 


266  PERMUTATIONS    AND    COMBINATIONS. 

Ex.  What  must  be  given  to  prt  in  a  life  of  10  years, 
when  the  ages  of  those  in  possession  are  40  and  50,  al-- 
lowing  6  per  cent,  for  money  ? 

Ans.  2.204,  or  not  quite  2i  years*  purchase. 

If  the  life  to  be  added  be  15  years,  the  answer  would 
be  2.067,  or  very  little  more  than  2  years*  purchase.  And, 

If  the  life  to  be  added  be  20  years,  the  answer  would 
be  1.908,  or  less  than  2  years*  purchase. 


PERMUTATIONS  AND  CONBINATIONS. 


The  Permutation  of  quantities  is  the  changing  or 
varying  the  order  of  things. 

The  Combination  of  quantities  is  the  shewing  how 
often  a  less  number  of  things  can  be  taken  out  of  a 
greater,  and  combined  together,  without  considering 
their  places,  or  the  order  in  wiiich  they  stand. 

Gase  I.  To  find  the  number  of  changes  that  can  be  made 
of  any  given  number  of  things,  all  different  from  each 
other. 

Rule.  Multiply  all  the  terms  one  into  another,  and 
the  last  product  will  be  the  number  of  changes  required. 

Ex.  1.  How  many  changes  can  be  rung  on  12  bells  ? 

1X2X3X4X5X6X7X8X9X10X11X121=  479,001,600. 

Ex.  2.  How  many  days  can  eight  persons  be  placed 
in  a  different  position  at  a  dinner  table  ? 

Answer,  40320. 


PERMUTATION    AND    COMBINATIONS.  267 

Case  TI.  Any  number  of  different  things  being  given, 
to  find  how  many  changes  may  be  made  out  of  them, 
by  taking  a  given  number  of  quantities  at  a  time. 

Rule.  Multiply  the  number  of  things  given,  by  itself 
less  1,  and  that  product  by  the  same  number  less  2,  dimin- 
isliing  each  succeeding  multiplier  by  an  unit,  till  there  are 
as  many  products,  except  one,  as  there  are  things  taken 
at  a  time  the  last  product  will  br  the  answer. 

Ex.  1.  How  muny  changes  can  be  rung  with  4  bells 
out  of  12  ? 


12X  12—1  X  12  —2  Xl2— 3  =  1^X11X10X9  =  11880. 

Ex.  2.  How  many  changes  can  be  rung  with  5  bells 
out  of  10?  Answer,  30240. 

Ex.  3.  What  number  of  words,  containing  each  6  let- 
ters, can  be  formed  out  of  the  24  letters  in  the  alphabet, 
supposing  any  6  to  form  a  word  ? 

Answer,  96909120. 

Case  HI.  To  find  the  combinations  of  a  less  number  of 
things  out  of  a  greater,  all  different. 

Rule.  Take  the  series  I,  2,  3,  4,  &c.  up  to  the  less 
number  of  things,  and  multiply  them  continually  to- 
gether for  a  divisor :  then  take  a  series  of  as  many  terms, 
decreasing  each  by  an  unity,  from  the  greater  number  of 
things,  and  multiply  them  continually  together  for  a 
dividend.  Divide  the  latter  product  by  the  former,  and 
the  quotient  will  be  the  answer. 

Ex.  1.  How  many  combinations  can  be  made  of  10 
things  out  of  100  .^^ 

1X3X3X4X5X6X7X8X9X10 
(the  number  to  be  taken  at  a  time)=»,628,800 
100X99  X98X97'X96X95X94X93X92X9  I 
(the  same  number  of  terms  taken  from  100) 
=62,815,650,955,529,472,000. 
6281565095529472000 

and ^  =  17310309456440. 

3628800 


268  EXCHANGE. 

Ex.  2.  How  many  combinations  can  be  made  of  3 
letters  out  of  the  24  letters  in  the  alphabet? 

Answer,  2024  combinations  required. 

Ex.  3*  A  club  of  21  persons  aj^reed  to  meet  weekly, 
five  at  a  time,  so  long-  as  they  could,  without  the  same 
live  persons  meeting  together,  how  long  would  the  club 
t'xist?  Answer,  391  years. 

t'ASE  IV.  To  find    the  compositions  of  any' number,  in 

sets  of  equal   number:^,  the   things  or  persons  them- 
selves being  different. 

Rule.  Multiply  the  number  of  things  in  every  set 
continually  together,  and  the  product  is  the  answer. 

Ex.  1.  There  are  three  parties  of  cricketters,  in  each 
eleven  men,  in  how  many  ways  can  11  of  them  be  cho- 
sen, one  out  of  each  ? 

Answer,  11X11x11  =  1331. 

Ex.  2.  In  how  many  ways  can  the  four  suits  of  cards 
b'e  taken,  four  at  a  time  ?  Ans.  28561. 

Ex.  3.  There  are  four  parties  of  whist  players ;  in  one 
there  are  G,  in  the  second,  5,  in  the  third  4,  and  in  the 
fourth  3  persons^  how  often  can  the  set  differ  with  these 
persons?  Ans;  S6o. 


EXCHANGE. 


By  Exchangf^.  is  meant  the  bartering,  or  exchanging,^ 
the  money  of  one  place  for  that  of  anotlier,  by  means  of 
an  instrument  in  writing,  called  a  bill  of  exchange. 

Exchanges  are  carried  on  by  merchants  and  bunkers 
all  over  Europe,  and  are  transacted  on  the  Royal  Ex- 
change of  London,  the  ilovai  Exchange  of  Dublin,  the 
Exchange  of  Amsterdam,  and  those  of  the  principal  ci- 
ties of  this  country  and  the  continent. 

When  an  exchange  is  mentioned  between  two  places, 
one  place  gives  a  determined  price,  to  receive  an  unde- 
termined one. 


EXCIIA'U.E.  '269 

The  deternuncd  price  is  called  certain  :  thus, 

London  i^iv6s  a  pound  sterling,  which  is  a  certain 
price,  lo  receive  from  Paris  a  number  of  francs,  more  or 
less,  to  he  paid  or  received  there.  As.i,ain  London  gives 
100'.  which  is  a  certain  price,  to  Duhiin  and  other  parts 
of  Ireland,  for  an  uncertain  number  of  pounds,  shillings 
anil  pence  Irish,  to  he  paid  or  received  there,  "viz.  frora 
105'.  to  115/.  Irish,  as  the  exchange  may  he. 

The  undetermined  price  is  called  uncertain,  because 
it  is  alvvays  8-ubject  to  variation  :  for  instance, 

London  pays  an  uncertain  price  to  Spain,  as  a  num- 
ber of  pence  sterling,  to  receive  a  dollar  which  is  cer- 
tain in  exchange. 

Thereat  money  of  a  state  signifies  one  piece  or  more, 
of  any  kind  of  metal  coined,  and  made  current  by  pub- 
lic authority,  as  guineas,  shilliiiiis,  &.c.  of  .^^ngiand. 

The  imaginary  moneif  is  chiefly  used  in  keeping  ac- 
counts, as  pounds  steiling,  for  which  there  is  no  coin  to 
nsTver. 

The  yar  of  exchange  is  the  quantity  of  the  money, 
whether  real  or  imaginary,  of  otie  countrv,  which  is 
equal  in  value  to  a  certain  quantity  of  the  money  ot 
another ;  thus^ 

200/.  sterling  is  equal  in  value  to  108/.  6<i.  Sd.  Irish: 
and  100/.  sterling  is  worth  IK)/,  of  the  currency  in  the 
West  Indies,  and  ecfual  to  16G/.  Ms.  ^d.  currency  of 
the  United  States. 

The  course  of  excltange,  is  the  value  a2;reed  upon  by 
merchants  and  othei  s  and  is  continually  fluctuating  above 
or  below  tlie  par  of  exchange,  accordi;-^  as  the  demand 
for  bills  is  greater  or  less. 

Jigio  denotes  the  dillerence  in  Amsterdam  and  other 
peaces,  between  current  money,  and  the  exchange  or 
bank-money,  the  latter  being  finer  than  tlie  former. 

Usance,  is  a  certUin  space  of  time  allowed  by  one 
country  to  another  for  the  payment  of  bills  of  exchange. 
Bills  are  either  payable  at  sight,  or  at  a  certain  number 
of  days  ^fter  sight:  at  usance,  double  umnce.  or  half 
usance.    At  oneytwo^  &-c.  usance  means  at  one,  two,  6^i'> 

S3* 


270  EXCHANGE. 

months*  date.    Half  usance  is  15  daySj  be  the  month- 
what  it  may. 

Bays  of  Grace  are  a  certain  number  of  days  allowed 
for  the  payment  of  bills  of  exchange,  after  the  expira- 
tion of  the  term  specified  in  such  bills,  and  are  variable 
in  diflferent  countries.  In  England  three  days  are  al- 
lowed. 

Rules  for  finding  what  quantity  of  the  money  of  one 
country  will  be  equal  to  a  given  quantity  of  the  money 
of  another  according  to  a  given  course  of  exchange. 

Case  I.  When  the  course  of  exchange  is  given,  how 
much  money  of  one  country  answers  to  a  certain  sum  of 
another,  as  of  Great  Britain  ? 

Rule.  As  the  given  course  of  exchange,  is  to  one 
pound  sterling,  so  is  the  given  sum  in  foreign  money,  to 
its  corresponding  value  in  sterling  money. 

Ex.  1.  How  much  sterling  money  can  I  have  for  2035 
Flemish  shillings,  when  the  course  of  exchange  is  37 
shillings  for  1/.  ? 
Here  1  say,  As  37  :  1  : :  2C35  ;  55  «=  pounds  sterl. 
Ex.  2.  How  much  sterling  money  can  1  get  for  4086 
florins,  4  stivers,  6  penings  banco,  supposing  1^.  is  worth 
38  schillings  and  2  grotes  r* 

schil.gr.         X.  florins  st.  p. 

38    2  1     :  :       4086    4     6 

12  40 


458  163440    grotes 

8     grotes  =  4  stivers 
I  of  a  grote=6  penings 

458)1 634481(356^.   I7s.  6d.  Ans. 

NOTE*  S.       d. 

*  8  penings    make  1  grote,  or  penny  =  0  0  54 

2  grotes         — —  1   stiver     -     -     -    =r  0  1.09 

12  grotes                 -  1  L'chilling     -    -    =  0     6  56 

20  schillings  1  pound  Flemish     =  10  11.18   ^ 

40  grotes       ^-—  1  guilder  or  Florin  =  1    9.8» 


EXCHANGE*  271 

Ex.  3.  What  sterling  money  will  293?.  10s.  6i.  Irish 
fetch,  when  the  exchange  is  1 14/.  Irish  for  100/.  sterling  ? 

114/-  :  lOOl,     :  :  293/.  10s.  Qd.  :  257/-  9s.  C^d. 

Ex.  4.  Dublin  remits  to  London  826/.  I3s.,  what 
must  be  received  there,  exchange  being  HO/,  per  cent? 

Answer,  751/.  10s. 

Ex.  5.  Jamaica  remits  to  London  287/.  Os-  lO^d, 
currency,  what  must  be  received  for  it,  exchange  being 
135/.  per  cent.  ? 

Answer,  212/.  ]2s.  5d, 

Case  II,  Given  the  course  of  exchange,  to  bring  any 
quantity  of  sterling  money  into  the  money  of  another 
country. 

Rule.  As  11.  sterling  is  to  the  course  of  exchange, 
so  is  the  given  sum,  in  sterling  money,  to  its  correspond- 
ing value  in  foreign  money. 

Ex.  1.  How  much  Flemish  money  will  233/.  6s.  8d» 
sterling  be  worth,  when  the  exchange  is  34s.  per  l/. 
sterling  ? 
1/.  ;  34s.  :  :  233/.  6s.  Si.  :  396/.  13s.  4i.  Answer. 

Ex.  2.  How  much  Flemish  money  must  be  given  for 
628/.  10s.  sterling  when  the  exchange  is  33s.  8c/.  per 
L,  sterling.  Answer,  1057/.  I9s*  6d, 

Case  III.  To  reduce  the  currency  of  any  state  into 
bank  or  exchange  money. 

Rule.  As  100,  with  the  agio  added  to  it,  is  to  100,  so 
is  any  given  sum  current  to  its  value  in  bank  money. 

Ex.  1.  How  much  bank  money  can  a  merchant  in 
Amsterdam  have  for  5550  guilders,  when  the  agio  is  4^ 
per  cent.  ? 

1044  :  loo  :  :  5550  :  6311  • Answer. 

•  104.5  . 


272  EXCHANGE. 

Ex.  2.  How  many  floiins  bank  will  3000  currency 
purchase,  agio  being  6i  per  cent.  ? 

Answer,  2H23  florins,  21  grotes,  I  penning. 

Case  IV.  To  reduce  bank  money  in<o  currency. 

Rule.   As  1 00  is  to  100,  with  the  agio  added  to  it,  so 
is  the  bank  money  to  the  currency. 

Ex.  1.  How  much  currency  can  I  have  in  Venice  for 
1500  ducats  bank,  when  the  aoiois  15  per  cent  ? 
100  :    115   :  :   1500  :   1725 

Ex.  2.   How  much  currency  can  1  have  for  5000  bank 
florins,  agio  being  8  per  cent. 

Answer,  5400  florins.- 


IRELAND. 


Account*  are  kept  in  Ireland  as  in  England,  viz.  in 
pounds,  shilling,  and  pence. 

The  par  of  excliange  in  Ireland  is  108Z.  6s.  8d.  ;  that 
is,  108/.  6^.  Sd.  Irish  is  equal  in  value  to  100/.  sterling  ; 
or  Is.  id.  Irishy  is  equal  to  one  shilling  English. 

The  course  of  Exchange  varies  from  1051.  to  115/. 
according  to  the  balance  of  trade.     See  page  269. 

Ex.  I.  London  remits  to  Dublin  300^.  sterling,  what 
must  be  received  for  it,  exchange  bein<2;  106/.  Irish  per 
cent.,  ami  also  when  it  is  1 12  per  cent.  ? 

Here  I  say.  As  100  s  106  :  :  SOO  :  318?.  Answer, 
and         100  :  IIS  :  :  300  :  356/.  Answer.  . 

Here  it  is  evident,  that  when  England  remits  the 
certain  price  ti?  another  country,  the  higher  the  ex- 
change, the  greater  advantage  is  derived  by  England  j 


27^ 

lor  when  the  exclKin^e  is  100,  she  will  receive  418/.  for 
her  3O0/.,  and  when  it  is  112/.,  she  will  receive  33GZ.  ibr 
the  same  sum. 

Ex.  2-  Duhlin  remits  to  London  700/.  Irish,  what  is 
it  eqijal  to  when  the  exchanj^e  is  108/.,  and  also  when  it 
is  110/.? 

Here  Dublin  remits  the  certaith  and  London  gives  the 
uncertain  price,  and  I  say, 

As   108  :   100  :  :  700  :  61-8/.  2^.  Hi/.  Answer, 
no  :   100  :  :  700  :  636/.  7s»     3il.   ^Answer. 

Here  Dublin  is  gainer  when  the  exchange  is  low,  be- 
cause, in  that  case  700/.  purchases  048/.  2s.  11^^.,  and 
in  the  other  it  purchases  only  630/.  7a'.  Sd. 

Ex,  3.  London  remits  to  Dublin  54-Sl.  lOs.  sterling, 
for  how  much  Irish  must  London  be  credited,  exchange 
being  11 C^  ? 

Answer,  602/.  15s.  6|-(L 

Ex.  4.  Dublin  remits  to  London  900/.  15s.,  how  much 
sterling  must  be  received,  exchange  bein^  112/.  ? 

Answer,  804/.  5s.  nearly. 
Ex.  5.  I  purchase  sundry  books  in  Dublin,  for  which 
I  give  as  follows  : 

For  the  first      -       -      -      L,  0     9 
second  -'     - 
third       -       -       - 
fourth    -       -       - 
what  are  they  worth  in  English  money  ?■ 

Answer,  2/.   ISs.  6|-??.  nearly. 


0  9     6") 

^  ^^     ^V.[rish 

1  5     0) 


AMERICAN  STATES. 


Ruj^R. — As  the  value  of  one  dollar  of  the  given  State 
currency,  is  to  the  value  of  one  dollar  of  the  required 
State  currency,  so  is  the  currency  given  in  the  question, 
to  the  sum  required. 


The  exghang-e  heing  at  par. 


27-2  EXCHANGE. 

EXAMPLES. 

Ex.  I.  How  much  Maryland  currency  must  I  have 
for  3500/.  of  New  York  currency  ? 

s.         s.    d.  L, 

8:76::     350O 
12  20 


90 

70000 
90 

8)()30O0OO 
12)787500 

2,0)6562,5 

Answer,  L,  3281  5 

Ex.  2.  In  576/.    105.  New  England  currency,    how 
much  South  Carolina  ?  Answer,  448/.  7s.  9^d. 

Ex.  3.  Bring  6274/.  5s.  South  Carolina  Currency,  to 
North  Carolina  currency  ?     Answer,    10755/.   17s.   ]^d. 

Ex.  4.  How  much  Canada  currency  in  5000/.  New 
York?  Answer,  3125/. 

Ex.  5.  In  464/.  7s.  8c?.  Pennsylvania  currency,   how 
much  South  Carolina  ?       Answer,  283/.  18s.  l\d,  ^\ 

Ex.  6.  In  694/.  13s.    4c/.    Maryland  currency,    how 
much  New  York  currency  ? 

Answer,  740/.  19s.  6|c?. 

Ex.  7.  Exchange  for  South  Carolina  currency  1000/. 
Canada  currency  ?  Answer.  933/.  Qs,  8d. 

Ex.  8.  How  much  Georgia  currency  in  426/.  12s,  4d. 
New  Jersey  ?  Answer,  265/.  9^.  Od,  ^-^ 


EXCHANGE.  275 

AMERICA  AND  THE  WEST  INDIES. 


Accounts  are  kept  in  these  places  as  in  England,  in 
pounds,  shillings,  and  pence. 

Ex.  1.  London  remits  to  Barbadoes  945/.  17s.  ster- 
ling, how  much  currency  will  this  amount  to,  when  the 
C-Jchange  is  140  currency  ? 

Answer,  1 324?.  3s.  9|i. 

Ex.  2.  Sir  Francis  Baring  writes  word,  that  he  had 
received  for  me  a  remittance  of  one  quarter's  dividend 
on  4000  dollars,  at  5^  per  cent,  interest  and  the  ex- 
change is  i'>4  per  cent,  what  has  he  to  pay  me  ? 

In  this  cast  the  regular  interest  is  55  dollars,  which  at 
4s.  6d.  each,  v^t^n  exchange  is  at  par,  or  at  l66l.  13s. 
4i.,  would  be   12L  ns.  6d.,  but  the  exchange  is  164  5 
therefore  I  say. 
As  164  :  166^.  13s.  4rf.  :  >  12^  7s.  6d.  :  l2l.  lis.  6^^.  Ans. 

Or  by  Aecimals, 
164  :  166.66,  &c.  : :  53  :  55.^^9  dollars  ZZ  12Z.  lis.  6.d 

The  following  is  a  Table  of  the  Course  of  Exchange^ 
taken  with  slight  variations  from  Hie  Monthly  Maga- 
zine for  the  J  St  of  May,  1808. 

COURSE  OF  EXCHANGE. 

April  5; 
gives     34.5 
34.7 

35.5.2U.- 
34.9 
23.13       - 


Hamburgh 
Altona    -     gives 
Amsterdam  gives 
Ditto,  sight  gives 
Paris,  l.d.  gives 
Leghorn  receives 
Naples      ditto 
Genoa      ditto 
Lisbon     ditto 
Oporto      ditto 
^ladrid     ditto, 
Palermo  ditto 
Dublin     ditto 
Agio  of  Bank  \ 
of  Holland  $ 


49J  pence- 
42  ditto  - 
45    ditto - 

60    ditto 60  ; 

65    ditto 65$ 

SH|  doEff- 


April  12, 
34  6  for  U. 
34  7  for  do 
35.4.2.U.  do 
34  8  for  do 
l.d. 24.0     for  do[rials 


49| 
42 


for  1  pezza  of  8 
for  1  ducat 
for  1  pezza 

for  1  milrea 

— -      for  1  dollar 


92  per  oz. — 
11 01/.         — 

6|  per  cent 


.  92 
.110 


per  oz. 
tor  100 


6|  percent. 


276  e:xckangk. 

Tliis  table,  in  addition  to  what  is  gone  before,  will 
afford  an  opportunity  of  explaining  everything  that  a 
mail  of  business  vvill  wish  to  be  aciiuainted  with. 

On  the  5th  of  April,  the  exchange  between  Hamburgh 
and  London  was  at  the  rate  of  34  schillings,  5  grotes  for 
a  pound  sterling;  that  is,  if  a  merchant  in  London  sell 
a  bill  on  Hamburgh  for  500/.,  he  would  be  paid  for  /t 
34.5  X  500  zz  17208  schillings,  4  grotes;  but  on  che 
12th,  such  bill  would  have  fetched  34.6  X  500  =  J/250 
schillings.  Here,  the  higher  the  exchange  the  greater  the 
advantage  to  England  ;  for  the  merchant,  in  this  in- 
stance, gains  41  schillings,  8  grotes,  by  the  ^ise  in  the 
exchange. 

For  Altona,  the  course  of  exchang-'^  ^s  the  same  on 
both  days,  viz.  the  L.  is  worth  34  schillings,  7  grotes  ; 
and  for  Amsterdam,  the  course  o^  exchange  falling,  the 
merchant  in  London  would  be  a  loser,  who  put  off  his 
market  from  the  oth  to  the  i-^th. 

In  this  case  S5.5  2U.  means,  that  a  pound  sterling  is 
worth,  on  the  5th  35  ^^chillings,  5'grotes,  allowing  it  to 
be  payable  at  two^ionths'  date  :  but  if  it  is  payable  at 
sight,  it  is  then  n^orth  only  34  schillings  9  grotes.  This 
difference,  w/iich  on  a  bill  of  100/.  is  equal  to  34  schil- 
lings 4  grotes,  is  instead  of  the  interest  of  money  for  the 
interval. 

The  course  of  exchange  rose  between  London  and 
Paris  from  the  5th  to  the  12th  of  April.  On  the  first  of 
these  days  1/.  uas  at  l.d.,  that  is,  at  one  day's  sight, 
worth  23.13,  or  23  francs,  and  13  cents.;  but  on  the 
12th  its  value  was  24  francs. 

Leghorn  receives  49j  pence  for  1  pezza  of  8  rials, 
that  is,  a  bill  of  exchange  of  5O0O  pezza  would  be  worth 
4s.  ^d.  multiplied  by  5000,  or  1036/.  9s.  2d,  A  Na- 
ples ducat  was  worth  3s.  6d.  :  a  Genoa  pezza  3s,  9di : 
a  milrea  of  Lisbon  5  shillings,  one  of  Oporto  5s.  5d. 


EXCHANGES.  277 

Madrid  receives  SS^d.  Eff.  for  1  piastre  of  8  rials,* 
that  is,  a  Spanish  piastre  of  exchange  was  worth  3s.  2|i. 

A  species  of  paper  money,  denominated  tvt/es  rials,  is 
circulated  in  Spain,  the  value  of  vvhich,  independently  of 
interest  on  them,  is  this: — Vales  rials  for  000  dollars 
are  worth  9035  rials,  10  maravedies  of  ve^/on,t  that  is,  as 
34  maravedies  is  equal  to  one  rial,  1  dollar  payable  in 
this  sort  of  paper  is  worth  15  ria4s,  2  maravedies.  The 
paper  is  transferable  by  indorsement ;  and,  by  lawi 
should  be  received  in  payment  according  to  the  nominal 
value;  but  as  it  experiences  depreciation,  it  is  necessa- 
ry in  drawing  on  Spain  for  effective  money,  to  insert  the 
words  '' payable  in  effective"  in  the  body  of  the  bill, 
which  might  otherwise  be  payable  in  vales  rials  :  hence 
the  word  Eff.  in  the  table,  which  is  an  abridgment  of 
"  in  effective'^*. 


NOTES. 

*  In  some  parts  of  Spain  they  reckon  by  silver  money, 
which  is  of  two  kinds,  viz.  old  and  new  plate,  the  former 
is  the  most  valuable  :  thus  the  piastre  of  exchange  con- 
sists of  8  rials  old  plate,  or  of  10  rials  new  plate,  the  rial 
being  at  the  par  of  exchange  worth  little  more  than  o\d, 

t  The  copper  money  of  Spain  is  called  vellon. 
In  Madrid,  and  theprincipal  places  ot  Spain,  accounts 
s^re  kept  in  piastres  (called  also  dollars)  rials,  and  mara- 
vedies 5  and  sometimes  in  ducats. 


TABLE. 


s.    d. 
51- 


34  maravedies   1  C   \  rial        li:  0    5 

8  rials  I   make  ■<    1   piastre  ZZ  3     7 

375  maravedies  J  (^   1  ducate  zz  4  Ijf 

Hence  the  piastre  at  par  is  3s.  Id.,  and  the  ducate  at 

par  4s.  Iji. ;  but  the  course  of  exchange  of  the  piastre 

varies  from  35  to  45  pence. 


278 


EXCHANGES/ 


Palermo  92  pence  per  cz.  In  Sicily  exchanges  are 
made  per  onza  by  the  ounce  of  Silver,  for  which  on  the 
day  referred  to  Palernao,  received  92  pence,  or  7s.  Sd.* 

Dublin  1 10^-  for  100/,  that  is,  at  the  date  of  the  table 
there  would  have  been  given  on  the  exchange  of  London 
a  bill  on  Dublin  for  110/.  5s.  for  100/.  sterling,  fcee 
page  275. 

By  the  agio  of  the  Bank  of  Holland  is  meant,  as  we 
have  seen,  page  269,  the  difference  between  cash  and 
bank  money,  which,  by  the  table,  is  on  the  5th  of  April, 
6|,  or  6/.  10s.  percent.  ;  that  is,  106/  lOs.  currency 
must  be  given  for  100/.  bank,  and  so  in  proportion. 

Exchange  between  London  and  other  Ftaces  in  this 
Country, 

The  several  cities,  towns,  &c.  in  Great  Britain,  ex- 
change with  London  for  a  small  premium  in  favour  of 
London,  as  from  ^  to  1  or  |A  percent.  The  premium 
is  more  or  less  according  to  the  greater  or  less  distance, 
and  according  to  the  demand  for  hills. 

Ex.  York  draws  on  London  for  560/.  10s.  exchange 
being  -J  per  cent. ;  how  much  money  must  be  paid  at 
York  for  the  bill  ? 


i 

?w 

560  10 

0 

-^ 

i 

2  16 

«i 

1     8 

oi 

L.564   14     0| 

To  avoid  paying  the  premium,  which  in  some  cases, 

would  not  be  just,  it  is  the  usual  practice  to  take  the  bill 

payable  a  certain  number  of  days  after  date.     On  this 

principle,  interest  being  5  per  cent,  73.  days  are  equiva- 

365 
lent  to  \L  per  cent  because  —  =73. 

5 

NOTE. 

*  The  Sicilian  ounce  is  600  grains,  and  the  monies  are 
regulated  by  the  following  Table  i 

10  grains       -      make      -     1  carlin, 
2  carlins      -      make      -     1  tarin, 
30  tarins       -    (600  gr.)  -    1  ounce. 
A  crown  (seudo)  is  equal  240  grs.,  therefore  5  crowns 
—2  ounces. 


EXCHANGE.  279 

Ex.  A  friend  at  Exeter  has  received  for  me  OS  sjuineus, 
in  which  lie  is  no  ways  interesteil,  and  haviuir  no  means 
of  sending  the  money  liut  by  a  bill  of  exchange,  he  a<^rees 
with  iiis  banker  to  draw  it  30  days  after  date,  rather 
than  pay  the  premium  of  |  per  cent.,  is  my  friend,  or  the 
banker,  tlie  gainer,  allowing  5  per  cent.  ? 

Answer,  the  banker  loses  Is.  2d.  of  his  usual  profit. 

EXAMPLES    FOR    PRACTICE. 

Ex.  1.  How  much  currency  will  6330  {guilders,  bank- 
money,  be  worth  iu  Holland,  agio, being  8^  per  cent.  ? 

Answer,  7176  guilders,  39  grotes/ 

Ex.  2.  What  is  the  agio  of  3310  guilders,  6i  per  cent.? 
Answer,  206  guilders,  S5  grotes. 

Ex.  3.  A  London  merchant  draws  on  Amsterdam  for 
1564/.  sterling  ;  how  many  pounds  Flemish,  and  how 
many  guilders  will  that  amount  to,  exchanyje  being  3 1 
schil.  8  gro.  per  L.  sterling.     See  table,  page  270. 

Answer,  2710  18    8=pounds  Flemish — 1G265     24 
guilders. 

Ex.  4.  How  much  sterlinu:  money  will  pay  a  Portu- 
guese bill  of  exchange  of  1654^=^372  'millreas  ;  that  is,  of 
1654  niillreas  and  372  reas,  exchange  being  65|^  pence 
sterling  per  millrea  ?*  Answer,  451/.  lOs.  l^i.  ■^^^. 

NOTE. 

*In  Portugal  accounts  are  kept  in  reas  and  millreas, 
the  latter  being  equal  to  1000  of  the  former  ;  and  they 
are  distinguished  from  each  other  by  some  such  mark  as 
that  in  the  question. 

The  millrea,  in  exchange  with  this  country,  is  at  par 
67 1  sterling  or  5s.  7|  sterling,  and  the  course  usually 
runs  from  5s.  3d.  to  5s.  Si. 

TABLE— Par  in  sterling.  s.  d.f. 

1   rea  IZ  0  0  0.27 

400  reas  >       l    S   ^  crusade       zz  2  3  0 
1000  reas  5  ^^^^  I   I  millrea        IZ  5  7  ^ 
The  reas  being  the  thousandth  parts  of  the  millreas,  are 
annexed  to  the  integer,  and  the  work  proceeds  as  in  de- 
cimals. 


52,30  EXeHANOE. 

Ex.  5.  How  many  Portuguese  reas  will  750?,  sterling 
amount  to,  exchange  being  64|  per  millrea  ? 

Answer,  ^2785  milr.  299  reasfff||. 

Ex.  6.  A  Spanish  merchant  imports  from  Seville, 
gomls  to  the  value  of  1031  piastres,  6  rials  :  how  much 
srerii Jig  money  will  this  amount  to,  exchange  being,  on 
the  day  of  payment,  41^  pence  per  piastre  ?"  See  Ta- 
ble, page  277.  ?  Ans.   187/.  Is,  a|^. 

Ex.  7.  I  want  to  purchase  goods  at  Cadiz,  and  for 
this  purpose  pay  into  a  Spanish  house  lOOOZ.  :  how  muek 
value,  in  piastres,  may  I  expect,  exchange  being  3s.  6^0?. 
per  piastre  ?  Answer,  5647^2/t  piastre^, 


ARBITRATION  OF  EXCHANGES, 

The  coarse  of  exchange,  between  nation  and  nation, 
naturally  rises  or  falls,  as  we  have  seen,  according  as 
the  circumstances  and  balance  of  trade  may  happen  to 
vary.  To  draw  upon,  and  ta  remit  money  to  foreign  pla- 
ces, in  this  fluctuating  state  of  exchange,  in  the  way  that 
will  turn  out  mo&t  profitable  is  the  design  of  arbitration. 

Arbitration  of  Exchange,  then,  is  a  method  of  finding 
such  a  rate  of  exchange  between  any  two  places,  as  shall 
be  in  proportion  with  the  rates  assigned  between  each  of 
them  and  a  third  place. 

By  comparing  the  par  of  exchange  thus  found,  with  the 
present  course  of  exchange,  a  person  is  enabled  to  find 
which  way  to  draw  bills  or  remit  the  same  to  most 
advantage. 

Arbitration  of  exchange,  is  eitheF  simple  or  compound. 

In  simple  arbitration,  the  rates  of  exchange  from  one 
place  to  two  others  are  given,  by  which  is  found  the 
correspondent  price  between  the  said  two  places,  called 
the  arbitrated  price. 

An  example  or  two  will  make  the  subject  clear. 


EXCHANGl^.  2^t 

Ex.  J.  If  exrhan;5e  between  Ti)nilon  ami  Amster- 
dam be  34  schil.  9  §rotes  per  L.  sterlini^.  and  if  excban<;e 
between  London  and  Genoa  be  45  pence  per  pezza 
what  is  the  par  of  arbitration  between  Amsterdam  and 
Genoa  : 

Here  IZ.  =24.0  pence:  therefore,  as 

240fl?.  :  34.S.  9  gr.  :  :  45i,  :  78-A^  g)\ 

Answer,  78  Flemish  grotto,  or  pence  per  pezza  Genoa 

Kx.  2.  If  exchange  from   London   to  Amsterdam 
3Ss.  9:/.  per  L  an<l  if  exchange  from  London  to  Paris  be 
32t/.  per  crown,  what  must  he  the  rate  of  exchange  from 
Amsterdam  to  Paris  r         Ans.  54d.  Flemish  per  crown. 

Ex.  3,  Tf  exchange  from  Paris  to  London  be  32(1  per 
crown,  and  if  exchange  from  Paris  to  Amsterdam  be  54d. 
Fleinish  per  crown,  what  must  b;^  the  rate  of  exchange 
between  London  and  Acnsterdam,  in  order  to  be  on  a  par 
with  the  other  two  ?  Answer,  S3  9  per  L, 

Ex.  4.  Amstertlam  exchanges  on  London  at  55  schil. 
5  grot«s  per  L.  sterlln  -  ;  and  the  exchanoje  between 
London  and  Lisbon  is  60  pence  per  millrea,  what  is  the 
exchange  between  Amsterdam  a»d  Lisbon  ? 

Ans.  J00.25grote9. 

The  conrse  of  exchange  being  given,  and  the  par  of 
arbitration  found,  we  obtain  a  metiiod  of  drawing  and 
remitting  to  advantage. 

Ex.  5,  If  exchange  from  L(mdon  to  Paris  be  32  pence 
sterling  per  crown,  and  to  Amsterdam  405  Flemish  per 
L..  and  if  I  learn  that  the  course  of  exchani>;e  between 
P.iris  and  Amsterdam  is  fallen  to  52  pence  Flemish  per 
crown  :  what  may  be  trained  per  cent.,  i)y  drawing  oa 
Paris  and  remitting  to  Amsterdam  } 
24^ 


282  COMPOUND    ARBITRATION. 

By  Ex.  2,  the  par  of  arbitration  between  Paris  and 

Amsterdam  is  54id.  Flemish  per  crown  ;  then 

d.      cr.        L.        cr. 

32  :  1  :  :   loo  :  750  drawn  at  Paris. 

cr.  d.Fl.       cr.        ^.Fi. 

1   :  52  :  :  750  :  39000  credit  at  Amsterdam. 

rf.Fl.  /..       rf.FI.       L.     s.   d, 

405  :   I  ::  39000  :  96     5   11  to  be  remitted. 

therefore  100^.— 96/.  5s.   lid.  =3/.  lis.   lfi.=gain  per 
cent.  to        r- 

If  the  course  of  exchange  between  Paris  and  Amster- 
H#am  he  at  56  Flemish  per  crown,  instead  of  52  ;  and  iff 
would  gain  by  the  negociation,  I  must  draw  on  Amster- 
dam  and  remit  to  Paps  ;  thus, 

X.    d.lX      Z.        f/.Fl. 
1   :  405  ::  lOO  :  40500  drawn  at  Amsterdam. 
d.¥\,  cr.        rf.FI.       cr. 
56  :   1   :  :  405OO  ;  723  credit  at  Paris, 
cr.     d.         cr.       L.     s. 
1   :  32  :  :  723  :  96    .8 
fherefore  100/.— 96/.  8s.  zz  3/.  12s.  gain  per  ceijl, 


COMPOUND  ARBITRATION. 


In  Compound  Arbitration,  the  rate  of  Exchange 
between  three  or  more  places  is  given,  to  find  how  much 
a  remittance  passing;  through  them  all  will  amount  to  at 
the  last  place  :  or  to  find  the  arbitrated  price,  or  par  of 
arbitration,  between  the  first  and  last  place. 

Examples  of  this  kind  may  be  worked  by  several  suc- 
cessive stagings  in  the  Rule  of  Three,  or  according  to  tlie 
following  Rules  i 


OOMPOttJND    ARBITRATION.  25» 

{l)  Distinguish  the  given  rates,  or  prices,  into  ante- 
cedents and  consequents,  placing  the  antecedents  in  one 
column,  and  the  consequents  in  another,  with  the  sign 
of  equality  between  them. 

(2)  The  first  antecedent,  and  the  last  consequent  to 
which  an  antecedent  is  required,  must  be  of  the  same 
kind. 

(3)  The  second  antecedent  must  be  of  the  same  kind 
with  the  first  consequent,  and  the  third  antecedent  of 
the  same  kind  with  the  second  consequent,  &c, 

(4)  Multiply  the  antecedents  together  for  a  divisorj 
and  the  consequents  together  for  a  dividend,  and  the 
quotient  will  be  the  answer  required. 

Kx.  If  a  merchant  in.  liondon  remit  500/.  sterling  to 
Spain  by  way  of  Holland,  at  35  shillings  Flemish  per* 
pound  sterling,  thence^o  France  at  58  pence  per  crown, 
thence  to  Venice  at  10  crowns  for  6  ducats,  and  thence 
to  Spain  at  360  mervadies  per  ducat  ;  how  many  pias- 
tres of  272  mervadies  will  the  500/.  amount  to  in  Spain  ? 

H.  =  35s.  or  4,Q0d.  FL 

5Sd.  =  1   crown 

10  cr.  ZZ  6  ducats 

1  due.  =       360  mervadies 

272  mer.  =3  I  piastre 

How  many  piastres  =500/. 

Omitting  the  units,  we  have  by  the  rule^ 
420  X  6  X  360  X  500 

.  and   this  fraction  reduced  to  it^ 

,      58  X  10  X  272 

21X3X45X500  ]  417500 

lowest   terms,  gives = =s= 

29  X  17  41)3 

2875|  piastres,  \^hich  is  the  answAff. 


1284 


DUODECIMALS. 


By  the  Rule  of  Three  we  should  hav6  said, 

1/.         :     420rf.         :  :     5i>0L  :     2IO000df. 

58^/.         :        1   cr.        :  ;     210()00c?.    ;     3620  cr.* 
10  cr.    :       6   due.    ":  :     3')20  or.     ;     2172  due. 
1   due.:     360  mer.,  :  :     2172  due  :     781Q20  mer. 
272  mer.:       1   pias.     :  :     78l920mer:     2875»  pias. 
If  the  course  of  direct  exchange  to  Spain   were  42| 
pence  sterling,  then  500/.  remitted   would  only  p.nount 
to  2823-^-    piastres,  of  course  2875|— 2823^,  gives    52, 
which  is  the  number  of  piastres  gained  by  the  negotiation.. 


DUODECIMALS 


Duodecimals,  or  Crnsa  jyiiiltlplicatian,  is  made  u^e 
of  by  artificers  in  measuring  theif  several  works,  and  is 
performeil  by  means  of  the  following  table  ; 

12""  fourths         -         make  1   third. 

12'"    thirds  -         -  1  second. 

12"     seconds       -         -  1   inch. 

12'      inches  -         -  1   foot. 

Glaziers,  Masons,  and  others,  measure  by  the  square 
foot. — Painters,  PavU.rs,  Plasterers,  &e  ,  by  the  square 
yard — elating,  tiling,  flooring,  &c.  by  the  square  of 
100  feet. — Brickwork  is  n^easured  by  the  rod  of  16|  feet, 
the  square  of  which  is  27-21. 

Rule,  (l)  Arrange  the  terms  of  the  multiplier  under 
the  same  denomination  of  the  multiplicand.  (2)  Multiply 
each  term  in  the  multiplicand,  beginning  at  the  lowest, 
by  the  (eet  in  the  multiplier,  aud  write  the  result  of  each 
under  its  respective  term,  observing  to  carry  one  for 
every  twelve.  (3)  Multiply,  in  the  same  manner,  by 
the  inches,  and  set  the  result  of  each  term  one  place  re- 


NOTE. 

•  The  fractions  are  omitted,  and  on  that  account  the  answer 
by  this  method  will  not  be  quite  accurate. 


DUODECIMAIS.  2^5 

moved  to  the  right-hand  of  those  in  the  multiplicand.* 
(4)  Multiply  then  by  the  seconds,  setting  the  result  of 
each  term  two  places  removed  to  the  right-hand  of  those 
in  the  multiplicand. 

Multiply  9  ft.  4  in.  8  sec.  by  5  ft.  8  in.  6  sec. 
9     4     8 
5     8     6 


46  11     4 

6    *3      1     4'" 
'4840' 


53     7      1     8     0 

Ex.  1.  How  much  must  I  pay  for  a  slab  of  marble 
7  ft.  4  in.  long,  and  2  ft.  1  in.  6  sec.  broad,  at  the  rate 
of  7s.  per  square  foot  ? 

Answer,  5L  9s.  id, 

Ex.  2.  What  will  ^e  the  expence  of  glass  for  a  win- 
dow that  measures,  in  the  clear  10  ft.  6^  in.  in  height^ 
and  4  ft.  9  in.  in  width,  at  is.  9d.  per  toot  ? 

Answer,  4?.  7s.  6d* 

Ex.  3.  How  much  will  a  room  cost  in  painting,  at 
Old.  per  yard  ;  the  sides  are  18  ft.  10  in.  by  10  ft.  3  in. 
and  the  two  ends  are  16  ft.  6  in.  by  10  ft.  3  in.  ? 

Answer,  3/.  3s.   8|-c?. 

Ex.  4.  What  shall  I  have  to  pay  for  statuary  marble 
about  my  fire-place,  at  14s.  per  foot ;  the  hearth  mea- 
sures 6  ft.  4  in.  by  2  (t.  3  in.,  the  three  fronts  are  each 
4  ft.  2  in.  by  8  in.,  and  the  mantle-piece  slab  is  6  ft  by 
9  in.  ?  Answer,  18/    19s. 

Ex.  5.  What  will  the  paving  of  a  court-yard  come  to, 
at  Is.  2d,  per  foot,  the  yard  being  74  feet  long,  and 
56ft.  Sin.  wide?  Answer,  244^.  12s.  ^d. 


*  Feet        multiplied  into  feet        give  feet. 
Feet         multiplied  into  inches     give  inches 
Feet         multiplied  into  seconds  give  seconds. 
Inches     multiplied  into  inches    give  seconds. 
Inches     multiplied  into  seconds  give  thirds. 
Seconds  multiplied  into  seconds  give  foufths. 


586  DUODECIMALS. 

Ex.  6.  How  much  shall  T  have  to  pay  for  slatins;  a 
house,  consistina;  of  two  slopinj^  sides,  each  nieasurinj^ 
24  ft.  5  in.  hy  15  ft.  9  in.  at  the  rate  of  41s.  per  square 
of  100  feet?  Answer,  loL   18s.  7d, 

Ex.  7.  What  will  the  tilinj^  of  10  houses  come  to,  the 
roof  of  each  house  consistin*^  of  two  sides,  each  18  feet 
by  14,  and  the  price  of  tiling  at  28s.  per  square  ? 

Answer,   701.  lis.  2ld. 

Ex.  8.  How  many  square  rods  are  there  in  a  brick 
wall  44  ft.  6  in.  long,  and  7  ft!  4  in.  high,  and  2^  bricks 
thick  ?*  Answer,  2  rods  nearly. 

Ex.  p.  If  an  oblong  garden  be  254  ft.  Bin.  long,  and 
184  ft.  8  in.  wide,  what  will  a  wall  cost  10  ft.  6  in.  high^ 
and  2J  bricks  thick,  at  15/.  15s.  per  square  rod  ? 

Answer,  888/.  6s. 

Ex.  10.  How  much  shall  I  have  to  pay  for  the  plate- 
glass  of  four  windows;  each  window  consists  of  16 
panes,  and  each  pane  measures  20|^  inches  by  15  J  inches 
at  9s.  6d,  per  foot  i  Answer,  t\Bl.  3s.  Sd. 


KOTE. 

*  Bricklayers  value  their  work  at  the  rate  of  a  brick 
and  a  half,  or  three  half  bricks  thick  ;  and  if  the  wall  be 
more  or  less  than  this,  it  must  be  reduced  to  that  thick- 
ness by  the  following;  rule  : — '<  Multiply  the  measure 
found  by  the  number  of  half  bricks,  and  divide  by  three  :*' 
thus,  if  the  wall  be  2^  bricks  thick,  1  multiply  by  5,  and 
divide  the  product  by  3. 

Ex.  If  the  wall  be  50  feet  long,  and  f  high,  and    2 

4  600 

bricks  thick,  it  will  be  50x9x— =600  feet  5  and  

3  272^ 

=  2^  square  rods  nearly. 


■^  if.' 


%^ 


